text:
Elizabeth Siddal

Elizabeth Eleanor Siddall (25 July 1829 – 11 February 1862), better known as Elizabeth Siddal (a spelling she adopted in 1853), was an English artist, art model, and poet. Siddal was perhaps the most significant of the female models who posed for the Pre-Raphaelite Brotherhood. Their ideas of female beauty were fundamentally influenced and personified by her. Walter Deverell and William Holman Hunt painted Siddal, and she was the model for John Everett Millais's famous painting Ophelia (1852). Early in her relationship with Dante Gabriel Rossetti, Siddal became his muse and exclusive model, and he portrayed her in almost all his early artwork depicting women.

Siddal became an artist in her own right and was the only woman to exhibit at an 1857 Pre-Raphaelite exhibition. Significant collections of her artworks can be found at Wightwick Manor and the Ashmolean Museum. Sickly and melancholic during the last decade of her life, Siddal died of a laudanum overdose in 1862 during her second year of marriage to Rossetti.

Early life
Elizabeth Eleanor Siddall, named after her mother, was born on 25 July 1829, at the family's home at 7   Charles Street, Hatton Garden, at the time in the parish of Saffron Hill, Hatton Garden, Ely Rents and Ely Place in central London. Her parents were Charles Crooke Siddall, and Elizabeth Eleanor Evans, from a family of English and Welsh descent. She had two older siblings, Ann and Charles Robert. At the time of her birth, her father had a cutlery-making business.

About 1831, the Siddall family moved to the less affluent borough of Southwark, in south London. The remainder of the Siddall children were born in Southwark: Lydia, to whom she was particularly close; Mary, Clara, James and Henry. Elizabeth Eleanor Siddall "received an ordinary education, conformable to her condition in life" and first "read Tennyson ... by finding one or two poems of his on a piece of paper" that had been wrapped around some butter. Literary analysts have noted that her artwork sometimes used subjects from Tennyson's writings and that his writings may have influenced her poetry.

Pre-Raphaelite model

In 1849, while working at a millinery in Cranbourne Alley, London, Siddal made the acquaintance of Walter Deverell. Accounts differ on the circumstances of their meeting. One account is that she became acquainted with Deverell's father, who worked at the Government School of Design, then at Somerset House. Siddal showed some of her artwork to him, and he introduced her to his son. In another account, William Allingham visited the milliner's to meet a woman he was acquainted with and admired; Siddal was the woman's co-worker and joined the pair on their walk home, as it was the women's usual practice to travel home from work together. Siddal made such an impression on Allingham that he recommended her as a possible model to his friend Deverell, who was struggling with a large oil painting based on the Shakespeare play Twelfth Night.

A third account has Deverell accompanying his mother to the millinery where he noticed Siddal in the back of the shop. In any case, Deverell later described Siddal as "magnificently tall, with a lovely figure, and a face of the most delicate and finished modelling   ... she has grey eyes, and her hair is like dazzling copper, and shimmers with luster." Deverell subsequently employed Siddal as a model and introduced her to the Pre-Raphaelites.

As with the other Pre-Raphaelites, Deverell took his inspiration directly from life rather than from an idealized classical figure. In his Twelfth Night painting, he based Orsino on himself, Feste on his friend Dante Gabriel Rossetti and Viola/Cesario on Siddal. This was the first time Siddal sat as a model. According to William Michael Rossetti, Dante Gabriel's brother, "Deverell drew another Viola from her, in an etching for The Germ." Elaine Shefer asserts that Deverell portrayed Siddal in A Pet and The Grey Parrot.

William Holman Hunt painted her in A Converted British Family Sheltering a Christian Missionary from the Persecution of the Druids (1849–1850) and Two Gentlemen of Verona, Valentine Rescuing Sylvia From Proteus (1850 or 1851).

For John Everett Millais's Ophelia, Siddal floated in a bathtub full of water to portray the drowning Ophelia. Millais painted daily through the winter, putting oil lamps under the tub to warm the water. On one occasion, the lamps went out and the water became icy cold. Millais, absorbed by his painting, did not notice and Siddal did not complain. After this, she became ill with a severe cold or pneumonia. Her father held Millais responsible and, under the threat of legal action, Millais paid her doctor's bills.

Siddal came to either embody or influence the Pre-Raphaelite ideals of feminine beauty.

Artwork and poetry

In 1853, Siddal signed The Lady of Shalott as "E. E. Siddal", the first time she had signed one of her works and an early instance of her shortened surname. By that same year, Rossetti had taken Siddal on as a student. He told his friend Ford Madox Brown that her "fecundity of invention and facility are quite wonderful, much greater than mine". Siddal seems to have inspired Rossetti, as he followed her in depicting the same subjects, and he reused her designs after her death.

Siddal's 1854 self-portrait (see right) diverged from the Pre-Raphaelites' typical idealised beauty. As Anna Solomon wrote, "she depicts herself looking harsher, angrier and less attractive than the languid Siddal of the Pre-Raphaelite paintings." From 1855 to 1857, art critic John Ruskin subsidised her career and paid £150 per year in exchange for all the drawings and paintings she produced. She produced many sketches, drawings, and watercolours as well as one oil painting. Her sketches are similar to other Pre-Raphaelite compositions illustrating Arthurian legend and other idealized medieval themes, and she was the only woman who exhibited with the Pre-Raphaelites at an 1857 exhibition at No. 4 Russell Place, Fitzroy Square, London. That same year, Siddal studied at the Sheffield School of Art.

During Siddal's career as an artist and poet from 1852 to 1861, she produced more than a hundred works. Unpublished during her lifetime, her poetry often dwelt on dark themes, lost love, or the impossibility of true love. Her small poetic output was nonetheless accomplished. Constance Hassett wrote that "Siddal's poetry ranges from the perfectly realized ballad narrative, to its opposite, the overheard lyric, and to something in between, the made-to-be heard monologue." Critic William Gaunt wrote that "Her verses were as simple and moving as ancient ballads; her drawings were as genuine in their medieval spirit as much more highly finished and competent works of Pre-Raphaelite art."

Relationship with Rossetti
Dante Gabriel Rossetti met Siddal in 1849, probably while they both modelled for Deverell. Rossetti gave Siddal the nickname "Lizzie" when she entered the Pre-Raphaelite Brotherhood circle, and "the diminutive enhanced her youthful, dependent role". By 1851 or 1852, they became engaged. Siddal had also become Rossetti's main model and muse, and he stopped her from modelling for others.

In 1852, Siddal began to study with Rossetti. She started staying at his Chatham Place residence, sometimes with him and sometimes by herself. They subsequently became anti-social and absorbed in each other's affections. They coined affectionate nicknames for one another, such as "Guggums" or "Gug" and "Dove", the latter one of Rossetti's names for Siddal. He also shortened the spelling of her surname to Siddal, dropping the second l.

During this period, Rossetti's most abundant and personal works were his pencil sketches of Siddal at home, most of which he entitled simply "Elizabeth Siddal". He portrayed Siddal in moments of leisure, such as reading, sitting, or in repose, or when painting or drawing. She also became the subject of much of Rossetti's poetry throughout their relationship and particularly after her death. Rossetti became obsessive in portraying Siddal. It has been estimated that there are thousands of Rossetti's drawings, paintings, and poems in which Siddal was a subject.

Beginning in 1853, Rossetti used Siddal as a model for a series of Dante-themed paintings, including The First Anniversary of the Death of Beatrice (1852), Beatrice Meeting Dante at a Marriage Feast, Denies him her Salutation (1851), Dante's Vision of Rachel and Leah (1855), and, perhaps his most famous portrait of her, Beata Beatrix (1864–1870), which he painted as a memorial after her death.

As Siddal came from a working-class family, Rossetti feared introducing her to his family. Siddal was the victim of harsh criticism from his sisters. The knowledge that his family would not approve contributed to Rossetti's delaying the marriage. Siddal appears to have believed, with some justification, that Rossetti was always seeking to replace her with a younger muse, which contributed to her later depressive periods and illness. Although Ruskin urged Rossetti to marry in 1855, their relationship deteriorated: the reasons probably included Siddal's ill-health, her laudanum addiction, Rossetti's philandering, Rossetti's lack of funds, the aforementioned disapproval of the Rossetti family, and Rossetti's probable aversion to marriage in general.

In 1857, Siddal gave up her stipend from Ruskin and went to Sheffield, the birth place of her father, to attend the school of art there. She moved in with her cousin's family. A son of this family, Willie Ibbett, proposed to her, but she indicated that she was already engaged. However, by mid-1858, Siddal and Rossetti appeared to be both done with their engagement and little is known about Siddal from that time until 1860. In Spring 1860, Siddal's family contacted Ruskin with the news that Siddal was gravely ill. Ruskin in turn informed Rossetti. Siddal was at the seaside resort of Hastings. In a change of heart, Rossetti hurried to her side that April with a marriage license. Shortly before their marriage, Rossetti produced a famous portrait of Siddal, Regina Cordium or The Queen of Hearts (1860). This painting is a close-up, vibrantly coloured depiction of Siddal.

Siddal and Rossetti married on Wednesday, 23 May 1860 at St. Clement's Church in Hastings. There were no family or friends present, only a couple of witnesses whom they had asked. When Siddal's health improved, they honeymooned in Paris and Boulogne in the latter half of 1860, then returned to the Chatham Place residence that they expanded into an adjoining house. Siddal became pregnant and appeared to be happier and healthier.

Ill health and death
"In Elizabeth Siddal's constitution there was a consumptive taint. This may, I suppose, have come from the father; for the mother was a healthy woman, living on til past ninety." It was thought that she suffered from tuberculosis, but some historians believe an intestinal disorder was more likely. Elbert Hubbard wrote that "She suffered much from neuralgia, and the laudanum taken to relieve the pain had grown into a necessity." Others suggest that she may have been anorexic or that her poor health was due to laudanum addiction or a combination of ailments.

Siddal travelled to Paris and Nice for several years for her health. At the time of her wedding, she was so frail and ill that she had to be carried to the church, despite it being a five-minute walk from where she was staying. She became severely depressed and her long illness gave her access to laudanum to which she became addicted. By 1861, Siddal became pregnant, which ended with the birth of a stillborn daughter. The stillbirth left Siddal with post-partum depression. By early 1862, she had become pregnant for a second time.

Siddal overdosed on laudanum on 10 February 1862. She, Rossetti, and his friend Algernon Charles Swinburne had dined together in a nearby hotel. After having taken Siddal home, Rossetti attended his weekly lecture at the Working Men's College. Upon returning home from teaching, Rossetti found Siddal unconscious in bed and could not revive her. The first doctor Rossetti called claimed that he was unable to save her, upon which Rossetti sent for another three doctors. A stomach pump was used, but to no avail. She died at 7:20   am on 11 February 1862 at their home at 14 Chatham Place. Her obituary noted that she "had expressed no wish to die, but quite the reverse. Indeed [sic] she contemplated going out of town in a day or two, and had ordered a new mantle [sic] which she intended to wear on the occasion." The coroner ruled her death as accidental; however, there are suggestions that Rossetti found a suicide note, with the words "Please look after Harry" (her invalid brother, who may have had a slight intellectual disability), supposedly "pinned   ... on the breast of her night-shirt." Consumed with grief and guilt Rossetti allegedly went to see Ford Madox Brown who is supposed to have instructed him to burn the note. Since suicide was illegal and considered immoral, it would have brought scandal on the family and barred Siddal from a Christian burial.

After Siddal's death

Siddal was buried with her father-in-law Gabriele on 17 February 1862 in the Rossetti family grave in the west side of Highgate Cemetery. Later burials in the same grave are her mother-in-law Frances Rossetti (1886), Christina Georgina Rossetti (1895), and William Michael Rossetti (1919).

In August 1869, Rossetti authorized Charles Howell to disinter her coffin to retrieve a handwritten book of Rossetti's poems, which he had laid beside her head before burial. With the aid of a Dr. Llewelyn Williams and two others, Howell accomplished this in October 1869. Dr. Williams subsequently disinfected the book. Rossetti then published the contents in Poems (1870).

These became part of Rossetti's sonnet sequence entitled The House of Life. This sequence contained the poem "Without Her", a reflection on life once love has departed.

"What of her glass without her? The blank grey There where the pool is blind of the moon's face. Her dress without her? The tossed empty space Of cloud-rack whence the moon has passed away. Her paths without her? Day's appointed sway Usurped by desolate night. Her pillowed place Without her? Tears, ah me! For love's good grace, And cold forgetfulness of night or day.

What of the heart without her? Nay, poor heart, Of thee what word remains ere speech be still? A wayfarer by barren ways and chill, Steep ways and weary, without her thou art, Where the long cloud, the long wood's counterpart, Sheds doubled up darkness up the labouring hill. — Dante Gabriel Rossetti, 'The House of Life', Ballads and Sonnets"

Legacy
Their home at 14 Chatham Place was demolished and is now covered by Blackfriars Station.

Exhibitions and collections
A retrospective of Siddal's work was curated by Jan Marsh in 1991 at the Ruskin Gallery in Sheffield.

Rosalie Glynn Grylls bought some of Siddal's works at auction in 1961. These works became part of Wightwick Manor, donated by her husband, Geoffrey Mander, and her to the National Trust. A 2018 exhibition, "Beyond Ophelia", curated by National Trust Assistant Curator Hannah Squire, ran for nine months and featured twelve artworks by Siddal and owned by the National Trust. Only the second solo exhibition of her work, the exhibition examined Siddal's career, artistic style, subject matter, and the recognition challenges she faced as a female artist.

Siddal was among the women featured in the 2019 Pre-Raphaelite Sisters exhibition at London's National Portrait Gallery.

In 2023, the Tate Gallery had an exhibition The Rossettis which included 17 of Siddal's works.

Literature
Writer and curator Jan Marsh wrote that those fascinated by Siddal included Swinburne, Oscar Wilde, and Arthur Symons. The artist and author Charles Ricketts confessed that "Oh, we have all, when young, been in love with Miss Siddal." With the emerging fields of psychology and sexology, a reevaluation of Siddal in fiction, poems, and biographies occurred in the 1920s and 1930s. She became regarded as "a morbid, hysterical, suicidal woman clinging to her virginity and angrily jealous of her rivals". By the mid-twentieth century, perceptions had changed again so that Siddal became "a Pre-Raphaelite groupie, a child of the 1950s and 60s pop culture." More recently, authors and biographers have reassessed Siddal. In this reexamination, she emerges as "partly a victim of masculine oppression and partly a rediscovered proto-feminist", and a rediscovery that includes "a determined effort to detach Elizabeth Siddal's story from that of Rossetti and the PRB, ... and present her with a biography of her own."

Along with Algernon Charles Swinburne, Siddal and Rossetti are the subjects of "How They Met Themselves", which is part of The Sandman series by Neil Gaiman, drawn by Michael Zulli, and published in Vertigo: Winter's Edge #3 (2000). In it, a dying Lizzie drugged with laudanum has a last dream or vision in which the trio takes a train trip to a forest "where they each would see their true love". This story bears the same title as a drawing and a painting by Rossetti that both depict Siddal.

Television
Rossetti's relationship with Siddal has been the subject of television dramas, notably Dante's Inferno (1967), by Ken Russell, in which she was played by Judith Paris and Rossetti by Oliver Reed; The Love School (1975) in which she was played by Patricia Quinn; and Desperate Romantics (2009) in which she was played by Amy Manson. She was portrayed in a recurring role by Hannah Onslow in the Paramount+ adaptation of Elizabeth Macneal's The Doll Factory (2023).

Art
Siddal is depicted on one of the plates in the Famous Women Dinner Service by Vanessa Bell and Duncan Grant, (1932–1934), commissioned by the art historian, Kenneth Clark.

The Delaware Art Museum hosted a 2022 exhibit of Holly Trostle Brigham's works inspired by and portraying Siddal in conjunction with its Pre-Raphaellite collection that includes works by and a Rossetti portrait of Siddal.

Drawings

 * The Lady of Shalott (n.d.), J.S. Maas Collection
 * Lovers listening to Music (1954), Wightwick Manor, West Midlands
 * Pippa Passes (1854), Ashmolean Museum, Oxford, England

Paintings

 * Self Portrait (1853–54) – oil, private collection
 * The Quest of the Holy Grail (1855) – watercolour, Tate Gallery, London
 * The Haunted Wood (1856) – watercolour, Tate Gallery, London
 * Lady Affixing a Pennant to a Knight's Spear (1856) – watercolour, Tate Gallery
 * Madonna and Child with an Angel (c. 1856) – watercolour, Delaware Art Museum, Wilmington, Delaware
 * Sir Patrick Spens (1856) – watercolour, Tate Gallery, London
 * ''Clerk Saunders (1857) – watercolour, Fitzwilliam Museum, Cambridge, England
 * Lady Clare (1857) – watercolour, private collection

title: Elizabeth Siddal


text:
Elvetham Heath

Elvetham Heath is a residential area and civil parish, containing 1,894 homes on a 311 acre site, just outside the north western boundary of the town of Fleet in the English county of Hampshire. Construction of Elvetham Heath began in the summer of 1999 and ended in June 2008. The population of the civil parish at the 2021 Census was 5,269.

Location
Elvetham Heath was built on reclaimed pine plantation/heathland between the M3 motorway and the London-Southampton railway line, 2 mi north of Fleet town centre. Formerly known as Railroad Heath, it previously belonged to the local Calthorpe Estate.

Elvetham Heath serves as a commuter area to local towns; Basingstoke some 12 mi to the west, Reading is 16 mi to the north and London is 37 mi to the northeast. The development is located precisely midway between Southampton and London. It is adjacent to the M3 motorway, 12 mi from the M4, and 14 mi from the M25. Fleet railway station is approximately 1.2 mi from the centre of the development.

Nature reserve
The district has a Local Nature Reserve, Elvetham Heath LNR, managed by Hart District Council.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Elvetham Heath


text:
Talk:Continuous function

Counterexample for the equivalence of two definitions for topological spaces
The article originally stated:

The same statement with open neighborhoods instead of neighborhoods would be:

The last sentence then say these two statements are equivalent. That is false, here is a counterexample:

Let $$(X,\mathcal{T}_X)$$ and $$(Y,\mathcal{T}_{Y})$$ be two topological spaces with $$X = \{a,b,c\}$$, $$\mathcal{T}_{X} = \{\emptyset, \{a,b,c\}, \{a\}\}$$, $$Y = \{w,y,z\}$$ and $$\mathcal{T}_{Y} = \{\emptyset, \{w,y,z\}, \{w,y\}\}$$.

Let $$f \colon X \rightarrow Y$$ be a function from the first topological space to the second defined by $$f(x) = \begin{cases} w, & \text{if } x = a \\ y, & \text{if } x = b \\ z, & \text{if } x = c \end{cases}$$.

The only two neighborhoods of $$f(a)$$ are $$\{w,y\}$$ and $$\{w,y,z\}$$. $$f^{-1}(\{w,y\}) = \{a,b\}$$ and $$f^{-1}(\{w,y,z\}) = \{a,b,c\}$$. Both $$\{a,b\}$$ and $$\{a,b,c\}$$ are neighborhoods of $$a$$ so according to the first definition the function $$f$$ is continuous at $$a$$.

Now, the only two open neighborhoods of $$f(a)$$ are also $$\{w,y\}$$ and $$\{w,y,z\}$$. $$f^{-1}(\{w,y\}) = \{a,b\}$$ and $$\{a,b\}$$ is not an open neighborhoods of $$a$$. Therefore, according to the second definition the function $$f$$ is NOT continuous at $$a$$.

Therefore, the two definitions can't be equivalent. Otherwise, the function $$f$$ would be continuous under the two definitions. Hence, I have deleted the sentence claiming that the two definitions are equivalent.

Pyrrhonist05 (talk) 17:42, 5 November 2017 (UTC)

 * True, the relevant neighborhoods of a need not be open, but relevant neighborhoods of f(a) may belong (WLOG) to any basis of the filter, in particular, be open. Thanks to Pyrrhonist05. Boris Tsirelson (talk) 21:12, 5 November 2017 (UTC)

I've moved this from the article. I had copy edited something similar out of the article several months ago. This appears to restate the definition trivially in several ways, but I don't think it makes the definition any easier to understand:

Also, we usually don't state things in the notation of symbolic logic on Wikipedia (note that we don't do this for the other definitions of continuity). So, I've removed it pending discussion. Sławomir Biały (talk) 14:11, 6 November 2017 (UTC)
 * I didn't reintroduced this paragraph because it made the definition clearer but because I think it gave valuable information. I agree with you that it don't help to understand the first definition but I find that the definitions it adds are not merely trivial restatements of the first one. If you already know what are neighborhoods systems and the fact that they are filters (and so upper sets), it's true that the definitions are not really involved. But if you don't really know about neighborhoods systems being filters I think it is not so clear that the definition can be restated this way and that they are equivalent. Also, the first time I stumbled across this section of the page, I didn't known neighborhoods systems, filter and upper sets and it makes me discover these concepts. Therefore, I find that there is an opportunities in this section to introduce theses concept to the reader. Additionally, you may say that because all theses definitions are all equivalent we could just give one and forgot about the others but as a mathematician you know that sometimes when we try to prove something we don't succeed by using the first definition but then by using only a subtly changed definitions we manage to do it. What I want to convey is that I think it is important to give several definitions, at least to let people know there isn't only one. At the same time, I'm not under the impression that this content clutter the section or make it less readable so I don't see any reason for not keeping it. As for the symbolic notation, we could just remove it if that is a problem. Pyrrhonist05 (talk) 18:36, 7 November 2017 (UTC)
 * Any thoughts on this ?
 * Any thoughts on this ?
 * Any thoughts on this ?

 * To avoid being guilty of wp:original research here, it would help to have a source. - DVdm (talk) 10:08, 8 November 2017 (UTC)
 * I agree with Sławomir that this text is not useful for understanding the definition. It is also unnecessarily too WP:TECHNICAL, by using terms (filter and upper set), which are unknown by most users that know of continuous functions. Also, the last sentence consists of replacing a sentence like "this object has this property" by "there is an object having this property". This is never useful, and is confusing here.
 * By the way, this section is not an alternative definition, but the standard definition of the continuity at a point. I'll thus move and rename this section. D.Lazard (talk) 10:40, 8 November 2017 (UTC)
 * I never said that the content I reintroduced helped to understand the first definition, I said that it added valuable information. In my opinion, the definitions added are not the same as the first and are not merely simple restatement of it either. As for WP:TECHNICAL I think there is point where we just can't avoid to be technical. Should we remove the definitions using the hyperreals or oscillation because it is too technical ? Off course no ! If someone doesn't know filters and upper sets, I think it is fine since he or she can still understand and use the definitions and understand what is continuity without them. Pyrrhonist05 (talk) 15:24, 8 November 2017 (UTC)

Image in section "Continuous functions between topological spaces" is right
The image is reproduced here for clarity and completeness: This image is referring to this definition of continuity at a point between two topological spaces:

This definition is far from being unusual, I googled "topological space continuous function" and I found these web pages and PDF on the first result page: Now, one could argue that they are self-published works, so here a (more difficult to find) published work mentioning this definition: Also $$U$$ is not chosen after $$V$$, a $$U$$ such that $$f(U) \subseteq V$$ should just exists for (every) $$V$$ (neighborhoods) and the image just show that by picturing one such $$U$$ and illustrating what $$f(U) \subseteq V$$ means. One could say that the picture doesn't shows every $$V$$ (neighborhoods) but it's impossible to put everything in an image. An image is here to illustrate something so that it is better understood by schematizing things. Therefore the image is correct and should be restored because an image is always welcome to illustrate a point. Pyrrhonist05 (talk) 18:29, 8 November 2017 (UTC)
 * , page 4 definition 4.6.
 * , page 1 definition 1.1.
 * , page 1 Theorem 18.1, 4th item.
 * , first paragraph

Incorrect definition of Gk continuity
The wiki claims that these are identical to their curvature continuity (Ck) counterparts, but they are not. The geometric continuity (Gk) conditions require continuity of the specified degree along their parameterization in addition to the continuity in the space in which they are embedded. See p. 185ff of Gerald Farin's "Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide", 2nd ed. for details.

 * You are right. Thank you. Now corrected. Boris Tsirelson (talk) 06:23, 19 January 2019 (UTC)

On the "Weierstrass and Jordan definition"
Shouldn't this be "uniformly continuous", and not just "continuous"? There is a discrepancy here between this page and the [| uniform continuity page]. On this page (continuous function), it is simply called "continuous," but on the other (uniform continuity), it says "The difference between uniform continuity, versus ordinary continuity at every point, is that in uniform continuity the value of δ depends only on ε and not on the point in the domain. " I think something should be cleared up, here. — Preceding unsigned comment added by 2603:9001:2902:1300:849B:38FF:720A:4408 (talk) 05:36, 24 September 2019 (UTC)
 * There is no discrepancy, as, here, $$x_0$$ is defined before $$\delta,$$ which implies that $$\delta$$ may depend on $$x_0.$$ D.Lazard (talk) 06:36, 24 September 2019 (UTC)

Definition in terms of limits of functions
I like this definition because its somewhat easier to check against, as the example of the signum function demonstrates. It also looks very similar to the definition of Scott continuity - in fact Scott continuity looks like a generalization of this definition of continuity. If this is in fact the case, could we state this connection somewhere? I imagine many people from a programming semantics background would be interested! Houseofwealth (talk) 01:11, 16 January 2020 (UTC)

Error in limit definition
Greetings maintainers of this page. In the section "Definition in terms of limits of functions" you give the standard definition as

$$\displaystyle \lim_{x \to c} f(x) = f(c)$$. This is correct. Two paras later you say:

"The formal definition of a limit implies that every function is continuous at every isolated point of its domain."

This does not follow. In fact in the Wikipedia article on limits of functions at https://en.wikipedia.org/wiki/Limit_of_a_function, in the section "More general subsets," it is explicitly stated that limits can only be defined at limit points. That's because if you try to define limits at isolated points, the limits are not unique. That latter point is not made explicitly in the article, but it's the reason that limits are only defined at limit points.

Therefore, continuity at isolated points does NOT follow from the limit definition; but rather only from the epsilon-delta definition.

99.171.179.102 (talk) 21:14, 20 February 2021 (UTC)

Please remove the "highschool definition"
"The most common and restrictive definition is that a function is continuous if it is continuous at all real numbers. In this case, the previous two examples are not continuous ..."

and:

"... but is not continuous over the domain $$\R$$ because it is undefined at x = 0"

What is this noobish nonsense? $$f(x)=1/x$$ is a continuous function PERIOD - A function cannot be (dis-)continuous "over the domain $$\R$$" if it is not defined at every $$x\in\R$$. If something is defined in a wrong way in highschool (because it may be too complicated otherwise, I guess) this should not result in a Wikipedia article containing said wrong definition (as viable).

An example that is perfectly analogous to the above case: "The function $$f(x)=x^2$$, $$x\in\R$$ is not a continuous function over the domain $$\Complex$$ because it is undefined for $$x\in\Complex/\R$$."

I really hope it is clear that this is nonsense. --Felix Tritschler (talk) 12:56, 10 December 2021 (UTC)
 * I have clarified the definition(s) of (global) continuity. In particular, it depends on the context whether $$f(x)=1/x$$ is considered as a continuous function. D.Lazard (talk) 16:12, 10 December 2021 (UTC)

 * I'll cite somebody from above on a similar example:"If someone imagines tan(x) to not be continuous, they have not quite got it".
 * A function can only be continuous or discontinuous at points of its domain. "(1/x) ... is discontinuous at $${\displaystyle x=0}$$" is wrong just the same way it is wrong that $$f(x)=x^2$$, $$x\in\R$$ is (not) continuous at e.g. $$2+i.$$
 * I know there are many who really want to cling to the idea of continuity of functions as whether they can be drawn without resetting the pen. But this is not in line with modern day mathematics. --Felix Tritschler (talk) 21:11, 12 December 2021 (UTC)
 * In the context of calculus, "function" means often "partial function", and the domain of a function is often unknown, or, at least, difficult to compute. For example, are you able to describe without a computer the domain of $$f(x)=1/(x^6-4x^5+3x^4-7x^3+x^2-1)$$? Another example is the complex function $$f(z)=1/\Gamma(z),$$ where $$\Gamma$$ denotes the gamma function; if you are able to describe its domain, congratulation, you will win the Millenium Prize for having solved the Riemann hypothesis. An opposite example is the function $f(x)=\frac{x^2-2x+1}{x^2-1},$ which is not defined for $$x=1,$$ but becomes continuous if one defines $$f(1)=0.$$
 * All these examples (but the last) are continuous in their domain, but cannot be prolongated in a function that is continuous in a larger domain. For this reason, modern mathematicians say that these functions are continuous except at the zeros of their denominators, and these zeros are called discontinuities or singularities, or, in these particular examples poles.
 * In summary, this is you who are not in line with modern mathematics (modern mathematics are mathematics of living mathematicians, not necessarily mathematics taught in elementary calsses). D.Lazard (talk) 22:37, 12 December 2021 (UTC)
 * I doubt that most modern mathematicians would agree with the statement "$$f(x)=1/x$$ ... is discontinuous at $${\displaystyle x=0}$$" simply because it is plain wrong. It does not agree with the definitions of continuity given in this very article (- all definitions use the value of the function at the respective point, i.e. it has to be in the domain). I also doubt they treat the terms "singularity" and "discontinuity" as equivalent, in general. And if in certain areas of mathematics where mostly functions, the domain of which is difficult to evaluate, are considered and because of that, the term "continuous function" is used differently, then that should warrant a subheading within this article where this is specifically discussed. Also, could you please point me to review articles or similar sources where the term is used in the way you described it? --Felix Tritschler (talk) 18:23, 21 December 2021 (UTC)
 * Ok, apparently there have been some changes to the article in favor of my argument, thank you. There is still:
 * 1. "$$f(x)=1/x$$ ... is discontinuous at $${\displaystyle x=0}$$" - see above (or below)

 * 2. On functions with singularities: "When they are continuous on their domain, one says, in some contexts, that they are continuous, although they are not continuous everywhere. In other contexts, mainly when one is interested with their behavior near the exceptional points, one says that they are discontinuous."
 * - Near poles these functions are continuous and, again, at the poles they are not defined, hence, neither continuous nor discontinuous.
 * And the first sentence (of 2.), I think, just adds confusion to the topic. Because it is spot on, let me repeat the quote from above:"If someone imagines tan(x) to not be continuous, they have not quite got it"

 * Continuity of functions is about wheter they "respect" the limes, not about drawing the graph of a function in one uninterrupted line. This needs to be a fundamental message of the article.
 * --Felix Tritschler (talk) 12:25, 8 January 2022 (UTC)
 * Since there has been no answer, can we/I please remove/change the parts of the article talking about discontinuities outside the domain of functions? This includes the graph of f(x)=1/x including the wrong statement "... but is discontinuous at $$x=0$$" - Or we could keep the graph but with a paragraph talking about the very common mistake of this and other functions with singularities being discontinuous there. E.g. the French article on this subject has such a paragraph. And, btw, neither the German, the French nor the Spanish article claim these functions have discontinuities. It is a widespread misconception and this article should help correcting it, not the opposite. --Felix Tritschler (talk) 15:14, 20 May 2022 (UTC)
 * One can find, in many reliable sources sentences such that It is not the role of Wikipedia to decide whether this formulation is wrong or not, but, as such a formulation is very common, Wikipedia must report it. I agree that the section deserves to be improved or clarified, and, as a first step in this direction, I have added a link to Partial function. As your peremptory assertions are unsourced, they are considered here as your own original research. So, please do not base edits on them, this is against one of the main Wikipedia policies (see WP:NOR). D.Lazard (talk) 16:55, 20 May 2022 (UTC)
 * One can find, in many reliable sources sentences such that It is not the role of Wikipedia to decide whether this formulation is wrong or not, but, as such a formulation is very common, Wikipedia must report it. I agree that the section deserves to be improved or clarified, and, as a first step in this direction, I have added a link to Partial function. As your peremptory assertions are unsourced, they are considered here as your own original research. So, please do not base edits on them, this is against one of the main Wikipedia policies (see WP:NOR). D.Lazard (talk) 16:55, 20 May 2022 (UTC)

 * What I said is not original research. I already mentioned the French article and found in Arens, Hettlich - Mathematik (2018), page 219, for exactly the same example function (translated from German):"One can only speak of continuity in places where a function is defined". And I asked before if you could point me to sources stating the opposite but there was no answer. So in which mathematics textbooks (or other reliable sources) is it written that "The function $$x\mapsto \frac 1x$$ is discontinuous at $$x=0$$" (or similar, e.g. different function with singularity, with the same meaning)? --Felix Tritschler (talk) 16:27, 24 May 2022 (UTC)
 * The article Zeros and poles starts with "In complex analysis (a branch of mathematics), a pole is a certain type of singularity of a function, nearby which the function behaves relatively regularly", and it is later said that 0 is a pole of $$1/x.$$ You can find similar formulations in every textbooks of complex analysis. So the fact that in some contexts 0 is a singularity of $$1/x$$ is a sourced fact. Your peremptory assertion that this is wrong is thus your own opinion that is not a consensus of the mathematical community. D.Lazard (talk) 18:17, 24 May 2022 (UTC)

 * Yes, but this was never in question. The question was whether a pole/singularity is a discontinuity. I say, it is not, you say it is. You said in the post before your last one, that:"One can find, in many reliable sources sentences such that " - Where are they? I offered 2 sources that this is wrong, regarding even exactly the same example. --Felix Tritschler (talk) 08:35, 27 May 2022 (UTC)

 * This keeps getting better. Now you added complex analysis in the section "Real functions"-->"Definition"- again, without source (and with typos/missing articles). And the Cauchy Prinicipal Value part, you added as well, does not at all show that functions with singularities are discontinuous there - this concept allows for integration over singularities - why is this in the introductory section of the article "Continuous Function"? The part on partial functions also does not argue for functions with poles being discontinuous there, is phrased vaguely and also has a typo: "In other contexts, mainly when one is interested WITH their behavior near the exceptional points, one says that they are discontinuous."--> (I said this above already, btw):"near the exceptional points" they are continuous anyways (!), it is nowhere to find in the article "Partial Functions" that they are discontinuous at the poles and, so far, there are no sources backing it. - And again, where are your "many reliable sources sentences such that "The function $$x\mapsto \frac 1x$$ is discontinuous at $$x=0.$$"? Even if you come up with them (which I doubt for this example), I already gave two sources saying that this is wrong and I'm quite sure I can bring more. --Felix Tritschler (talk) 09:15, 2 June 2022 (UTC)

 * Again, there is no answer and no sources for your claim, so, so far, this is original research. I have provided two sources backing my position. So I repeat what I wrote above:"Can we/I please remove/change the parts of the article talking about discontinuities outside the domain of functions? This includes the graph of f(x)=1/x including the wrong statement "... but is discontinuous at $$x=0$$" - Or we could keep the graph but with a paragraph talking about the very common mistake of this and other functions with singularities being discontinuous there. E.g. the French article on this subject has such a paragraph. And, btw, neither the German, the French nor the Spanish article claim these functions have discontinuities. It is a widespread misconception and this article should help correcting it, not the opposite".
 * In addition, the complex analysis-part you added in the section "Real functions" should be removed for an obvious reason. --Felix Tritschler (talk) 12:23, 17 July 2022 (UTC)

Umm, how to put it politely: that's not correct. $$f(x)=1/x$$ is a standard textbook example of a function that is not continuous at x=0. It appears in pretty much all standard introductory pre-calculus textbooks used in high-schools throughout the land. Its a standard example, precisely because, to the young student, it looks like a nice smooth function, and so whenever some kid starts getting glib about continuity, you ask them to do a delta-epsilon proof at x=0, and they can't. Try it yourself. The entire raison d'etre of theorems and proofs is that, by being forced to write a detailed proof, you avoid glib but incorrect claims about this-and-such being true. You have to actually prove it. Try it. You'll fail. There is no proof of continuity at x=0. 67.198.37.16 (talk) 06:25, 29 December 2023 (UTC)

 * You are mistaken. For the epsilon-delta criterion (or the test for continuity via limits) at a position, the function has to be defined there. This is not the case for $$f(x)=1/x$$ at x=0 (and poles of functions in general). I went over this numerous times above - did you read that at all? Note that D.Lazard, in an attempt to justify his erratic notion of continuity, produced citation 7 (currently): Strang, Gilbert (1991). Calculus, which says:
 * "Objection: The definition makes $$f(x)=1/x$$ a continuous function! It is not defined at x=0, so its continuity can't fail. The logic requires us to accept this, but we don't have to like it."
 * Isn't that interesting? He, himself, gave a source that directly contradicted his view. Further, it's not about whether "we like it" or not, as it says in the source. It's about what is correct. And correct is that a function is continuous if it is continuous at every position of its domain - which is the case for $$f(x)=1/x$$. And a discontinuous function has at least one discontinuity - but it can only be discontinuous where it is defined. It's part of the definitions of both the limits- and the delta-epsilon-criterion. If some school- and undergrad books ignore these definitions and arrive at the conclusion that $$f(x)=1/x$$ is a discontinuous function, then they are wrong in this regard, which (being wrong) isn't that special for such books.
 * Btw I found that D.Lazard also wrote inaccurate content in the Wikipedia article on the square root, where he implies that the square root of e.g. 4 is {-2, 2}. He also contributed to the article "Multa-valued function", which doesn't exist in Wikipedias of other languages - simply because it doesn't exist at all. Apparently not enough mathmaticians/scientists who care for accurate content care enough for it to contribute to Wikipedia pages. This is very problematic because Wikpedia has such a wide reach. --Felix Tritschler (talk) 00:21, 12 April 2024 (UTC)
 * Please just use direct language instead of sarcasm or rhetorical flourishes, and try to stay polite and assume good faith. –jacobolus (t) 01:27, 12 April 2024 (UTC)
 * Could you please point me to where was I less polite and more sarcastic than "Umm, how to put it politely:"?
 * And regarding assuming good faith: In the original dispute from 2021/2022 with D.Lazard, he just stopped replying to my arguments that he could not counter and ignored the source I gave to back my standpoint - and his own source (Strang, Gilbert (1991)) contradicts his own standpoint. Yet, he didn't revert his changes (e.g. where now there is complex number content under the heading "Real Functions") to the article. --Felix Tritschler (talk) 11:03, 12 April 2024 (UTC)
 * I stopped replying, because I had replied before, and I do not like to waste my time to repeat the same thing again and again. About self-reverting my edits, I would do it if there were a consensus for that. In this case, you alone is certainly not sufficient for a consensus. D.Lazard (talk) 11:41, 12 April 2024 (UTC)
 * Yes, you replied, but you didn't address any of my objections. So it had the same effect as no reply. --Felix Tritschler (talk) 11:03, 15 April 2024 (UTC)
 * Parts such as "did you read that at all?", "an attempt to justify his erratic notion", "Isn't that interesting?", and "Apparently not enough mathmaticians/scientists who care for accurate content care enough for it to contribute to Wikipedia pages." seem entirely gratuitous to me. –jacobolus (t) 12:26, 12 April 2024 (UTC)
 * More productively, if you want to find sources, one keyword to try is "infinite discontinuity", which will turn up many textbooks, papers, university course notes, etc. (though sometimes this means something else). –jacobolus (t) 13:00, 12 April 2024 (UTC)

Suggestion to change the article based on the discussion 'Please remove the "highschool definition" '
Since there has been no reaction, specifically no sources backing that functions are discontinuous at singularities etc. (see above), I am suggesting here anew to change the article accordingly, that is, to clarify that functions are neither continuous nor discontinuous at singularities or at any other points outside their domain and, hence, that functions like $$x\mapsto \frac 1x$$ or the tangent function are continuous functions (A function that is continuous at every point in its domain is a continuous function). In the above discussion I have given a source from a mathematics textbook (and the French article on continuous functions). Btw, these issues with the article have been addressed before, in the now archived discussion "Continuity and the domain of the function" where it is correctly stated that "elementary functions are continuous" but either the article was never changed in this regard or these changes were reverted. (Plus, there have been very questionable recent additions and changes to the article like content on complex analysis in the section "Real functions".) I would encourage other users, especially the contributors of the now archived discussion, to get involved. Thank you in advance. --Felix Tritschler (talk) 11:50, 6 August 2022 (UTC)

 * You are proposing something that is contradicted by pretty much every textbook on pre-calculus used throughout the land. The reason this is a "high-school definition" is because it is used in high-schools. It is what students see, it is what is taught by convention. It appears on exams and in homework problems. 67.198.37.16 (talk) 06:31, 29 December 2023 (UTC)
 * Please have a look at my reply to your other comment (IP address 67.198.37.16 from 29.Dec.2023). --Felix Tritschler (talk) 11:05, 12 April 2024 (UTC)

Definition in terms of neighborhoods has issues
The definition begins with: "A neighborhood of a point c is a set that contains, at least, all points within some fixed distance of c." This is true for a metric space but there are topological spaces that are not metrizable.

The definition continues: "Intuitively, a function is continuous at a point c if the range of f over the neighborhood of c shrinks to a single point as the width of the neighborhood around c shrinks to zero." Again, the "width" is only well defined in a metric space.

It ends with "This definition only requires that the domain and the codomain are topological spaces and is thus the most general definition." This may be the case if we replace "neighborhood" by "open set" but given that neighborhoods are introduced as "points within some fixed distance", it is plainly wrong. 120.51.141.38 (talk) 05:51, 17 April 2023 (UTC)

 * As this section is a subsection of, there is no problem, except in the last paragraph, which I have fixed by mentioning that this is a generalization of the concept of neighborhood that is used there. D.Lazard (talk) 09:01, 17 April 2023 (UTC)

1/x is discontinuous at 0?
I know that some calculus textbooks does say that $$x\mapsto 1/x$$ is discontinuous at 0 since it is not defined there. But this is a kind of old way to describe the matter and the map $$x\mapsto 1/x$$ becomes continuous (even meromorphic) after defining $$1/0=\infty$$. If we are to keep this example in the article I think it's best to replace the function by a map
 * $$f(x)=

\begin{cases} 1/x&x\ne0\\ 0&x=0 \end{cases} $$ which is also defined at 0. 慈居 (talk) 08:42, 28 October 2023 (UTC)

 * And by the way, I'm here because this confusing statement has arrived at Korean Wikipedia by translation. 慈居 (talk) 08:47, 28 October 2023 (UTC)
 * The article says . I think that this answers your concern. Also, when not otherwise specified, $$x\mapsto 1/x$$ is a (partial) function from the reals to the reals or from the complexes to the complexes. Its extension as a function on the projectively extended real line or the Riemann sphere is continuous everywhere, but it is another function. D.Lazard (talk) 12:11, 28 October 2023 (UTC)
 * Oh, that definition for partial functions is even a special case of continuous functions in the normal sense, by adding a new isolated point to the codomain. This is indeed enough to make the statement make sense. But as for my home wiki, I'd rather not introduce this definition and only discuss continuity at points in the domain, cause that seems enough. 慈居 (talk) 13:48, 28 October 2023 (UTC)
 * First, I doubt that the consensus is that, when not otherwise specified, $$x\mapsto 1/x$$ is assumed to be a partial function. But even if, that does not produce a discontinuity at x=0: Neither via the epsilon delta criterion, nor via limits is it possible to determine a discontinuity at x=0. As a partial function, x=0 still does not belong to the domain of this partial function and the criteria for discontinuity can only be applied at points of the domain of a (partial) function.
 * Please have a look at the source "Strang, Gilbert (1991). Calculus. SIAM. p. 702. ISBN 0961408820", where it says:
 * "The definition makes f(x)=1/x a continuous function! It is not defined at x=0, so its continuity can't fail." --Felix Tritschler (talk) 11:33, 12 April 2024 (UTC)
 * The meaning of the expression $x \mapsto 1/x$, without further explicit detail, depends on context and is somewhat fuzzy and not always consistent. In complex analysis, dealing with arbitrary rational functions, it is most convenient to be working with the projectively extended complex numbers, $\hat\C = \C \cup \{ \infty \}$, so naming this function $f$ we have $f(0) = \infty$ and is continuous there. In applications working with real-variable calculus, it's most common in practice to consider partial functions from $\R \to \R$, so that $f(0)$ is considered undefined and is a discontinuity because $\textstyle \lim_{x \to 0^+} f(x) \neq \lim_{x \to 0^-} f(x)$. In computer floating point arithmetic the number system includes signed zero (representing either 0 or possibly underflow) and signed infinity (representing either infinite quantities or overflow), so $f(0) = \infty$ and $f(-0) = -\infty$, both well defined but clearly a discontinuity. Some authors, especially of introductory textbooks, work only with functions per se, with the domain defined to exclude values where the function is undefined. For these authors, $f(0)$ is undefined and $\textstyle \lim_{x \to 0^+} f(x) = +\infty$, $\textstyle \lim_{x \to 0^-} f(x) = -\infty$, but the function's domain is said to be $(-\infty,0)\cup (0,\infty)$ (as is the function's image) and the function is said to be continuous for points in this (itself non-continuous) domain. –jacobolus (t) 17:19, 12 April 2024 (UTC)
 * As a real valued function of a real variable, isn't $$1/x$$ most commonly a function $$\mathbb R\setminus\{0\}\to\mathbb R$$? 慈居 (talk) 17:41, 12 April 2024 (UTC)
 * My impression is that defining continuity at an undefined point is rather rare. And the only reference that I know that does so, defines every function $$\mathbb R\setminus\{0\}\to\mathbb R$$ to be discontinuous at 0, because it is not defined at 0, which has nothing to do with analysis at all. 慈居 (talk) 17:58, 12 April 2024 (UTC)
 * Continuity has to do with whether the limit from the left is the same as the limit from the right and whether these equal the value of the function. In the case where the function is undefined at a particular point, it is indeed usually taken to be discontinuous there, but sometimes there is a removable singularity which can be trivially fixed by just defining the function at that point to be equal to its limit (e.g. starting from $x \mapsto \sin x / x$ we can explicitly define the value at $0 \vphantom)$ to obtain the continuous sinc function even though ordinarily division by zero is undefined), while other times there is some kind of "jump" where the limits from right and left do not agree, or a place where the function "blows up" and tends to infinity (from one or both sides, with the same or differing signs from above and below), or even an essential singularity where the behavior is more pathological. –jacobolus (t) 20:39, 12 April 2024 (UTC)
 * Such classification is made because we often consider a function from and to a subset of $$\mathbb C$$, but we also wish to see their behavior as a map from and to a subset of $$\widehat\mathbb C$$. But actually there is only one concept of "holomorphy" for a fixed domain and codomain. Holomorphy is defined for points in (the interior of) the domain and I think continuity should be too. 慈居 (talk) 05:53, 13 April 2024 (UTC)
 * In the original source which introduced the name "holomorphic function", Briot & Bouquet's Théorie des fonctions elliptiques (2nd edition, 1875), it is taken for granted that we can take the reciprocal of a function which has a pole at a point and get a zero there, or vice versa. That is, the pole of the function is at least implicitly considered to be part of the domain (or something close enough to the domain that it can counts as part of the domain of the reciprocal function). –jacobolus (t) 06:09, 13 April 2024 (UTC)
 * I see it but I didn't get what your point is. Another way to say this would be that a function holomorphic at points except some isolated poles has an analytic continuation to a domain that contains those poles, but valued in $$\widehat\mathbb C$$. The reciprocal function also has a continuation to a domain containing the infinity, and the two maps can be composed freely. 慈居 (talk) 06:23, 13 April 2024 (UTC)
 * ++ I'm fine with assuming that the domain already contains the poles, or not assuming it, which varies among different authors. I'm only concerned with defining continuity only at points in the domain. 慈居 (talk) 06:47, 13 April 2024 (UTC)
 * The current article text seems quite clear and explicit, discussing both points of view. Are you just wondering about finding sources for each version? For example, a 2021 version of introductory calculus notes from Harvard have "The function $f(x) = 1/x$ is continuous except at $x = 0$. It is a prototype with a pole discontinuity at $x = 0$. One can draw a vertical asymptote. The division by zero kills continuity." By comparison a 2023 version of introductory calculus lecture notes from Columbia have "However, this new definition doesn’t necessarily correlate with our intuition of which functions are continuous. For example, think about the function $f(x) = \tfrac1x$. This clearly has a discontinuity at $0$: the limit $\textstyle \lim_{x\to0} \tfrac1x$ does not exist, and neither does $f(0)$, so we certainly can’t say that they’re equal if they’re not even defined. However, at every other point f(x) is continuous, and since 0 is not in its domain this means f(x) is continuous everywhere in its domain and therefore continuous!" –jacobolus (t) 07:10, 13 April 2024 (UTC)
 * Uhm, I'm fine with discussing both points of view, but to me the current article looks "A, but some authors B" and I prefer "B, but some A". 慈居 (talk) 08:21, 13 April 2024 (UTC)
 * Repeating D.Lazard from above, the current article says explicitly:
 * "Many commonly encountered functions are partial functions that have a domain formed by all real numbers, except some isolated points. Examples are the functions $x \mapsto \frac {1}{x}$ and $x\mapsto \tan x.$ When they are continuous on their domain, one says, in some contexts, that they are continuous, although they are not continuous everywhere. In other contexts, mainly when one is interested in their behavior near the exceptional points, one says they are discontinuous."
 * What would be your preferred wording? –jacobolus (t) 16:01, 13 April 2024 (UTC)
 * If I may try (but in better English than mine): Many commonly encountered functions have a domain formed by all real numbers, except some isolated points. Examples are the functions $$\textstyle x\mapsto\frac1x$$ and $$x\mapsto\tan x$$. They are continuous functions, since each of them is continuous at every point in the domain. They are not continuous nor discontinous at the exceptional points, since those points do not belong to the domain. In other contexts, mainly when one is interested in their behavior near the exceptional points, one says that they are discontinuous at those points, and therefore not continuous functions on the entire real line. 慈居 (talk) 16:20, 13 April 2024 (UTC)
 * I second that, except for the second part ("In other contexts ..."), which is what currently is in the article, vaguely phrased (when is one "interested in their behavior near exceptional points"?), and directly contradicting the very definition of discontinuity (e.g. for the epsilon-delta criterion, the point in question has to be part of the domain) and also some of the sources and the source I gave from a maths textbook (see above, Arens, Hettlich - Mathematik (2018), page 219), plus the French and German Wikipedia on this topic --Felix Tritschler (talk) 11:14, 15 April 2024 (UTC)
 * One of my reasons is that this should be at least more compatible with the definition for topological spaces. There is no apparent natural parent spaces asggined to arbitrary spaces, so continuity of maps between spaces is only defined at points in the domain. 慈居 (talk) 08:27, 13 April 2024 (UTC)
 * This is gold. According to all variants you named except one, $x \mapsto 1/x$ is still a continuous function - and via the only (niche and most of all, inaccurate) one that makes $x \mapsto 1/x$ a discontinuous function, e.g. $x \mapsto 1/x^2$ would be a continuous function, using the same criteria. Wow! Also, treating $x \mapsto 1/x$ as a (non-total) partial function, there is no discontinuity at x=0, because, and I'm repeating myself here, x=0 is not part of the domain, so no criterion for discontinuity is applicable there. Treating it as a partial function doesn't change that in the slightest and it shouldn't matter whether there are authors who simply ignore this fact. Well, I'm out of here, both actively and passively, and you can even ban my account, I don't care any more.
 * I aimed at improving this article by employing the logical rigor actually required (and special) to mathematics, and provided the according sources, but it had no effect and nobody came to my support (maybe because a professor is involved? Also, some editors with the same cause already gave up, which can be seen in the archives). Of course there are sources with less rigorous standards, so there are controversial or even self-contradictory terms like 'infinite discontinuity' or "multivalued function" to be found on the internet or even in papers and books. In our lab, we used to often say jokingly: "Well, it's published, therefor it has to be true (laughs)" - There are high-quality sources, low-quality sources and everything in between. So far, I was under the impression that the auto-correcting forces within Wikipedia, and especially for important and basic/essential mathematics articles like this one, would naturally shape them guided by sources that are to the largest extent of high quality, used by editors who try to work towards the best possible encyclopedic entry, instead of to push through their own, non-mainstream, ideas.
 * But this is clearly not the case. Editors who have a personal interest in popularizing their own research and ideas, that are sometimes not even consistent with definitions in the same article they end up with, force these onto the masses by largely rewriting fundamental English Wikipedia articles like this one (the article 'Function (mathematics)' is another example) in a clearly biased fashion and with obvious issues, and ensuing valid criticism (not merely mine) on the talk-page is then answered with circular arguments (see talk-page of 'Multivalued function'), non-arguments (e.g. see my initial discussion with D.Lazard, where, in one response he described what 'poles' are, which was never in question and didn't help his argumentation at all), sarcasm ("congratulation, you will win the Millenium Prize for having solved the Riemann hypothesis" - but only I am later called out for sarcasm) or the assertion that many reliable sources support the case, followed by the production of a citation that actually does the opposite! Pointing all of this out is simply ignored. Then the article is outright vandalized by adding almost completely unrelated complex number content in the section 'Real Functions', which only I objected to and nobody took action against for almost two years. That alone should be an indication for a lack of editors (not readers!) still caring for this article.
 * In summary, none of this matters because the conflict is just "waited out" and for several reasons nobody jumps in to notice that the criticisms are actually valid, haven't been addressed by the author, and thus the article should be reverted (plus other clarifications added). Realizing recently how many other articles have been "improved" recently, I deem the situation unsalvageable and concede this fight against windmills. Btw, I really didn't expect to find mechanisms at work here that I would have a contextualized mostly with politics.
 * Finally, as it is now, because this is the Wikipedia with the widest reach, masses of students and otherwise interested individuals learn here, every day, that the square root of a positive real number y is usually taken to be both the positive and the negative solution of the equation y=x^2, that partial functions are almost as important as (total) functions in mathematics in general (even though they are a rather niche concept and mostly relevant in computer science, D.Lazard's field of research) and that e.g. rational functions are usually considered discontinuous functions, even though the definitions in the same article show this is plain wrong, no matter whether one considers them partial or total functions or uses extensions to the real numbers as functional values. No arguments, neither mine nor from 慈居 had an impact on the situation (they withdrew their interest being content with saving the Wikipedia article in their language). Note that, with my arguments and citations at hand, I could have, according to the rules, just made changes to the article myself, but I wanted to avoid an edit war and I was honestly hoping up to now that finally several other voices, e.g. the ones that argued in favor of my case a few years ago (part of which gave up, similarly to how I do now), would make themselves heard, so any changes in favor of a rigorously logical version would be enduring.
 * I don't expect an answer, and I wouldn't read it. --Felix Tritschler (talk) 12:32, 16 April 2024 (UTC)
 * Just for the record, repeats this again and again although there is no mention of complex numbers in . Should we deduce that he criticizes the article and its editors without having read it? D.Lazard (talk) 15:56, 16 April 2024 (UTC)
 * Can we all try to stay polite, please, even in the face of rudeness?
 * Felix Tritschler: being gratuitously condescending isn't selling your argument. Nor are sarcasm, hyperbole, mischaracterization of the article and other editors' comments, or bizarre false accusations.
 * I was under the impression that the auto-correcting forces within Wikipedia [...] would naturally shape them – Sorry to shatter your illusion, but no, this has never been true. Wikipedia has always been, and remains, full of disagreements/disputes, contradictions, frustration, and huge amounts of work remaining. –jacobolus (t) 16:36, 16 April 2024 (UTC)
 * No arguments, neither mine nor from 慈居 had an impact on the situation. The situation was in the progress! It takes time for the community to reach a consensus. This will only slow it down. 慈居 (talk) 16:54, 16 April 2024 (UTC)
 * Then perhaps better use the standard definition (as in Starng's Calculus) by default in the article? As for the other one, we could also mention that, citing a reference; I've seen such a definition, but not in an English reference, so I'm not going to add one myself. 慈居 (talk) 17:37, 12 April 2024 (UTC)

In both the real number system and the complex number system, infinity is not a number. In particular, the real number system obeys the Archimedean axiom. which says that for any two positive real numbers, some integer multiple of the smaller is bigger than the larger. If infinity were allowed as a real number, the Archimedean axiom would not hold.

What is called calculus at a lower level is, at a higher level, called either Real Analysis or Complex Analysis, the name indicating which number system is used. Rick Norwood (talk) 11:32, 28 October 2023 (UTC)

title: Talk:Continuous function


text:
User:Joe Canuck/ban

Date: Fri, 20 Jun 2003 12:40:58 -0700 From: Jimmy Wales  Subject: [WikiEN-l] Joe Canuck To: wikien-l@wikipedia.org

Joe Canuck is banned from wikipedia. Would someone with the appropriate powers please take the appropriate steps to make this technically true.

As always, Joe is invited to write to me to discuss this further, and/or is invited to the mailing list.

Some have expressed a strong conviction that Joe Canuck is the same person or persons as DW, etc. I express no opinion on that matter, and find that Canuck's actions alone are sufficient for banning. To the extent that it is true, of course, then just being the same person is grounds for a continuation of the ban.

--Jimbo

The above mailing list post is archived here.

Jimmy Wales wikien-l@wikipedia.org Fri, 27 Jun 2003 12:17:58 -0700

"Wikipedians should generally refrain from witch hunts of users who may be reincarnations of banned users."

However, grave concerns have been raised, and I have reason to think, based on what's been going on the past few days, that ChuckM, Joe Canuck, DW/Black Widow, and perhaps some others may (or may not) be the same person, posting repeatedly on the same sorts of topics and in the same sorts of obnoxious ways.

So, after a bit of research by one of the sysops, who will remain nameless unless he wants to take public credit, we've determined that there is a relatively small block of ip numbers where this user is coming from. At least through the weekend, we're going to block all of those completely.

Next Monday morning, we'll assess whether it worked, and discuss when to unblock that range of addresses. It seems unfortunate in case people in a certain ip block in Canada happen to want to edit this weekend, but odds are, they won't.

Additionally, any and all of those names can be banned by names, and I authorize a witch hunt (but please, practice WikiLove) on new instances. :-) Just for the weekend.

My concern is that this might spiral out of control into a full-fledged attack, and I'd just as soon prevent that and deal with it next week.

--Jimbo

 * Above post archived here

title: User:Joe Canuck/ban


text:
Gartner

Gartner, Inc. is an American technological research and consulting firm based in Stamford, Connecticut, that conducts research on technology and shares this research both through private consulting as well as executive programs and conferences. Its clients include large corporations, government agencies, technology companies, and investment firms.

In 2018, the company's client base consisted of over 12,000 organizations in over 100 countries. As of 2022, Gartner has over 15,000 employees located in over 100 offices worldwide. It is a member of the S&P 500.

History
Gideon Gartner founded Gartner, Inc. in 1979. Originally private, the company launched publicly as Gartner Group in 1986 before Saatchi & Saatchi acquired it in 1988.

In 1990, Gartner Group was acquired by some of its executives, including Gartner himself, with funding from Bain Capital and Dun & Bradstreet. The company went public again in 1993. In 2000, the name was simplified from Gartner Group to Gartner.

In 2000, Gartner coined the term Supranet. Gene Hall has been the CEO of the company since August 2004.

In the course of its growth, Gartner has acquired numerous companies providing related services, including Real Decisions—which became Gartner Measurement, now part of Gartner's consulting division—and Gartner Dataquest, a market research firm. It acquired Datapro Information Services, a computer industry analysis-focused unit of McGraw-Hill, in 1997; that in turn had once been Datapro Research Corporation of Delran Township, New Jersey.

It has also acquired a number of direct competitors: Meta Group in 2005, AMR Research and Burton Group in early 2010, and Ideas International in 2012.

In 2013, whilst working for Gartner, Anton Chuvakin coined the term "endpoint threat detection and response" for "tools primarily focused on detecting and investigating suspicious activities (and traces of such) other problems on hosts/endpoints".

In March 2014, Gartner announced that it had acquired the privately held company Software Advice for an undisclosed amount.

In 2014, Gartner also coined the term "Digital BizOps" and further developed the early philosophy for digital business operations. In July 2015, Gartner acquired Nubera, the business app discovery network that owns properties like GetApp (a peer review site), AppStorm, AppAppeal, and CloudWork. Terms of the deal were not disclosed. In September 2015, it acquired the privately held peer review site (PRS) Capterra.

In June 2016, Gartner announced that it had acquired the privately held company SCM World, headquartered in London, U.K. On January 5, 2017 Gartner announced it had reached an agreement to acquire CEB, Inc. in a cash and stock deal worth about US$2.6 billion. On March 7, 2017 Gartner announced that it has agreed to buy New York–based L2 Inc, which specialises in benchmarking the digital performance of brands. Terms of the deal were not disclosed.

title: Gartner


text:
Lucille Hegamin

Lucille Nelson Hegamin (November 29, 1894 – March 1, 1970) was an American singer and entertainer and an early African-American blues recording artist.

Life and career
Lucille Nelson was born in Macon, Georgia, the daughter of John and Minnie Nelson. From an early age she sang in local church choirs and theatre programs. By the age of 15 she was touring the US South with the Leonard Harper Minstrel Stock Company. In 1914 she settled in Chicago, Illinois, where, often billed as "The Georgia Peach", she worked with Tony Jackson and Jelly Roll Morton before marrying the pianist-composer Bill Hegamin. She later told a biographer, "I was a cabaret artist in those days, and never had to play theatres, and I sang everything from blues to popular songs, in a jazz style. I think I can say without bragging that I made the 'St. Louis Blues' popular in Chicago; this was one of my feature numbers."

The Hegamins moved to Los Angeles, California, in 1918, then to New York City the following year. Bill Hegamin led the band accompanying his wife, the Blue Flame Syncopators; Jimmy Wade was a member of this ensemble.

In November 1920, Hegamin became the second African-American blues singer to record, after Mamie Smith. Hegamin made a series of recordings for Arto Records and then Paramount in 1922. One of her biggest hits was "Arkansas Blues", recorded for Arto and released on many other labels, including Black Swan. She recorded one of Tom Delaney's earliest compositions, "Jazz Me Blues", in 1921, and it went on to become a jazz standard. She subsequently played theatre dates but did not tour extensively.

Lucille Hegamin lived at the Shuffle Inn in Harlem from November 1921 to January 1922. On January 20, 1922, she competed in a blues singing contest with Daisy Martin, Alice Leslie Carter and Trixie Smith at the Fifteenth Infantry's First Band Concert and Dance in New York City. Hegamin placed second to Smith in the contest, which was held at the Manhattan Casino. Then from February to May of that year she toured with the African-American musical revue Shuffle Along and this was the second of three companies. In the first company Florence Mills had the same role with the same musical revue.

From 1922 through late 1926 she recorded over forty sides for Cameo Records; in this association she was billed as "The Cameo Girl". After her marriage to Bill Hegamin ended in 1923, her most frequent accompanist was the pianist J. Cyrill Fullerton. In 1926, she recorded with Clarence Williams's band for the Columbia label. She sang with a band that was led by George "Doc" Hyder in 1927 for a show in Philadelphia. Further into the decade she performed in further revues with Hyder that were staged in Harlem theaters.

She performed in Williams's Revue at the Lincoln Theater in New York and then in various revues in New York and Atlantic City, New Jersey, through 1934. In 1929 she performed on the radio program Negro Achievement Hour, on WABC, in New York. In 1932 she recorded two sides for Okeh Records.

About 1934, she retired from music as a profession and worked as a nurse. She came out of retirement in 1961 to record four songs, accompanied by a band led by Willie "The Lion" Smith, on the album Songs We Taught Your Mother, for Bluesville Records. In 1962 she recorded Basket of Blues for Spivey Records. She performed at a benefit concert for Mamie Smith at the Celebrity Club in New York City in 1964.

Hegamin died in Harlem Hospital, in New York City, on March 1, 1970, and was interred in the Cemetery of the Evergreens, in Brooklyn, New York.

Style
Hegamin's stylistic influences included Annette Hanshaw and Ruth Etting. According to Derrick Stewart-Baxter, "Lucille's clear, rich voice, with its perfect diction, and its jazz feeling, was well in the vaudeville tradition, and her repertoire was wide." Like Mamie Smith, Hegamin sang classic female blues in a lighter style, more influenced by pop tunes, than the rougher rural-style blues singers such as Ma Rainey and Bessie Smith, who became more popular a few years later.

title: Lucille Hegamin


text:
Santos, São Paulo

Santos (, Saints), officially Municipality of Estancia Balneária de Santos is a municipality in the Brazilian state of São Paulo, founded in 1546 by the Portuguese nobleman Brás Cubas. It is located mostly on the island of São Vicente, which harbors both the city of Santos and the city of São Vicente, and partially on the mainland. It is the main city in the metropolitan region of Baixada Santista. The population is 433,656 (2020 est.) in an area of 280.67 sqkm. The city is home to the Coffee Museum, where world coffee prices were once negotiated. There is also a football memorial, dedicated to the city's greatest players, which includes Pelé, who spent the majority of his career with Santos Futebol Clube. Its beachfront garden, 5,335 m in length, figures in Guinness World Records as the largest beachfront garden in the world.

Early colonization
There are reports about the island of São Vicente just two years after the official discovery of Brazil, in 1502, with the expedition of Amerigo Vespucci to explore the Brazilian coast. When passing through the island formerly named Goiaó (or Guaiaó) by the natives, the expedition decided to give it the name of Sao Vincente, for the day's saint. However, in 1531, due to the decline of the Portuguese crown's business in India, Brazil rose on importance. King D. João III sent for a squad for the demarcation of territories on the island of São Vicente. The captain, Martim Afonso de Sousa, discovered a small village and a dock, known as Porto de São Vicente. One of the exiles brought by Amerigo Vespucci's expedition, Cosme Fernandes, had founded the trading village, which had boomed. Miguel Alfonso took the town by force, granting land on the island to settlers. In 1543, with the completion of the construction of a chapel on a hillock in honor of Santa Catarina by Luís de Góis, Brás Cubas ordered the port to be moved to the site of Enguaguaçu, which was calmer. The town booked to facilitate the trade that was unlocked with this move. The Portuguese nobleman ordered the construction of Brazil's second, and at the time only hospital, as Hospital da Santa Casa de Misericórdia is closed, similar to the Santa Casa de Lisboa. The hospital was called Santa Casa de Misericórdia de Todos os Santos in Olinda was closed. The new town of Enguaguaçu was then known as the town of Todos os Santos. There is speculation that the name Santos would come from the port of Santos in Lisbon, similar to the location of the new settlement. Hence, the region close to Outeiro was known as "Vila do Porto de Santos", and later, just "Santos".

Twentieth Century
The export of coffee from the Port of Santos gave rise to the city and mostly accounted for the wealth of the city at the turn of the 20th century. Export and import through its port have made it the modern city one finds today and turned it into the indispensable outlet for the production of the powerhouse that is São Paulo State. Adorning the landscape of the port city are the canals that are over a hundred years old. In 1899, Santos was the point of entry for the bubonic plague into Brazil. In 1924 it became the seat of the Roman Catholic Diocese of Santos.

In October 2006, light crude oil was discovered off the coast in the Santos basin.

Santos became a tourist city from the 1910s onwards, with the construction of the International Hotel and Parque Balneário and the construction of the beach front gardens in 1935. To this day, tourism in Santos is one of the main economic activities, mainly linked to beaches and historical heritage.

Geography
Santos is about 50 km (31 mi) from the metropolis São Paulo, capital of the state São Paulo, which is also the most populous city in Brazil.

The municipality contains the Laje de Santos Marine State Park, created in 1993, the first marine park to be created by the state. It is divided into two distinct geographic areas: the heavily urbanized island and the continental area, about 70% of which is protected. The areas differ radically in terms of population, economy and geography.

Insular area
Santos partially lies on the island of São Vicente (Saint Vincent), whose territory is divided with the neighboring municipality of São Vicente. It is a densely urbanized area of 39.4 km² that houses almost all the inhabitants of the city. It includes a flat area - Plain Coastal extension of the State of São Paulo - which has altitudes that rarely go above twenty meters above sea level, and an area composed of isolated hills called the Mass of São Vicente, the former home and endowed an urban illegal occupation with a mix of families characterized by high and low incomes, whose height does not exceed 200 meters above sea level.

The flat region of the island is almost completely devoid of native vegetation, although in the north region of the island - especially in the Alemoa, Chico de Paula and Saboó neighbourhoods - there are still remnants of mangroves. Before the occupation of the area of the island by 'chácaras' - rural residences, and subsequently by urbanization, there was a vast flooded land covered by mangroves, the native Atlantic Forest, and coastal vegetation.

On the city hills one can still find vast areas covered by the native Atlantic Forest, in spite of the existing chácaras and banana harvesting farms in the area. The 'Lagoa da Saudade' (Homesickness Lagoon), a pond located in one of the aforementioned hills, Morro Nova Cintra, was known to host a kind of caiman. The lagoon is also a popular destination among families in the city due to its playgrounds, barbecue kiosks, picnic spots and green areas. The disordered occupation of the hills represents both an environmental as well as a geological risk: the deforestation leads to frequent landslides, mainly from January to March, the traditional rainy season in the region.

Most rivers in the island were channeled when engineer Saturnino de Brito designed the system of canals in the city. As examples, we can cite the rivers Dois Rios ("Two Rivers") and Ribeirão dos Soldados ("Soldiers Creek"), which is nowadays referred by santistas as the 'Canal 4' on Avenue Siqueira Campos.

Major water courses cut the island in the north, such as the Rio de São Jorge (St. George River), which suffers from the problems of pollution and silting due to the occupation of its banks by slums.

Beaches

 * José Menino Beach
 * Pompeia Beach
 * Gonzaga Beach
 * Boqueirão Beach
 * Embaré Beach
 * Aparecida Beach
 * Ponta da Praia Beach

Islands

 * Urubuqueçaba Island
 * Barnabé Island
 * Diana Island

Climate
Despite the fact that it is located just outside the tropics, Santos has a tropical rainforest climate (Köppen: Af) with no real dry season. Tropical rainforest climates are typically found near the equator, so Santos featuring this type of climate is an exceptional situation. All months of the year averages more than 60 mm of rainfall during the course of the year. Santos features warm weather throughout the year, though June in Santos is somewhat cooler (and drier) than January. Mean temperatures in the city are around 19 °C during wintertime and around 25 °C in the summer months. Precipitation in Santos is very high, amounting to around 2000 mm annually. Santos lies in one of the few isolated regions of Brazil outside of the tropical Amazon Basin that receive more than 2000 mm of total average precipitation annually, although nearby Ubatuba, approximately 140 km to the east-northeast, is considerably wetter than Santos, receiving an average of 2645 mm of precipitation annually.

Economy
The Port of Santos is the biggest seaport in Latin America, which handled 96 million tons and 2.7 million TEUs in 2010. It has large industrial complexes and shipping centers, which handle a large portion of the world's coffee exports, as well as a number of other Brazilian exports including steel, oil, cars, oranges, bananas and cotton.

As of 2014, the municipality of Santos was the 6th largest exporting city, by value, in Brazil by trading $4.36B (USD) worth of goods. The top four products exported from Santos were raw sugar (23% of total exports), refined petroleum (16%), coffee (15%), and soybeans (13%).

Airport and Air Force Base
Santos Air Force Base - BAST, a base of the Brazilian Air Force, is located in the adjoining city of Guarujá.

The city will be served by Guarujá Civil Metropolitan Aerodrome, located in Guarujá.

Media
In telecommunications, the city was served by Companhia Telefônica Brasileira until 1973, when it began to be served by Telecomunicações de São Paulo. In July 1998, this company was acquired by Telefónica, which adopted the Vivo brand in 2012.

The company is currently an operator of cell phones, fixed lines, internet (fiber optics/4G) and television (satellite and cable).

Notable people

 * Gilmar (1930-2013), former Santos FC football player and two-time FIFA World Cup champion (1958 and 1962) with Brazil national football team
 * Pepe (1935-), former Santos FC football player and two-time FIFA World Cup champion (1958 and 1962) with Brazil national football team
 * Affonso Giaffone (1968-), former racing driver
 * Diego Higa (1997-), drifting driver and Hyperdrive champion

Twin towns – sister cities

Santos is twinned with:

 * 🇨🇷 Alajuela, Costa Rica
 * Ansião, Portugal
 * Arouca, Portugal
 * 🇪🇸 Cádiz, Spain
 * 🇵🇪 Callao, Peru
 * Coimbra, Portugal
 * 🇵🇦 Colón, Panama
 * 🇷🇴 Constanța, Romania
 * Fernando de la Mora, Paraguay
 * Funchal, Portugal
 * 🇨🇺 Havana, Cuba
 * 🇲🇦 Kenitra, Morocco
 * 🇯🇵 Nagasaki, Japan
 * 🇨🇳 Ningbo, China
 * Porto, Portugal
 * 🇨🇳 Rizhao, China
 * 🇯🇵 Shimonoseki, Japan
 * 🇨🇳 Taizhou, China
 * 🇮🇹 Trieste, Italy
 * 🇰🇷 Ulsan, South Korea
 * 🇦🇷 Ushuaia, Argentina
 * 🇲🇽 Veracruz, Mexico
 * Viseu, Portugal

title: Santos, São Paulo


text:
National Physical Laboratory (United Kingdom)

The National Physical Laboratory (NPL) is the national measurement standards laboratory of the United Kingdom. It sets and maintains physical standards for British industry.

Founded in 1900, the NPL is one of the oldest metrology institutes in the world. Research and development work at the laboratory has contributed to the advancement of many disciplines of science, including the development of early computers in the late 1940s and 1950s, construction of the first accurate atomic clock in 1955, and the invention and first implementation of packet switching in the 1960s, which is today one of the fundamental technologies of the Internet. The former heads of NPL include many individuals who were pillars of the British scientific establishment.

NPL is based at Bushy Park in Teddington, west London. It is managed under the Department for Science, Innovation and Technology and is one of the most extensive government laboratories in the United Kingdom.

Precursors
In the 19th century, the Kew Observatory was run by self-funded devotees of science. In the early 1850s, the observatory began charging fees for testing meteorological instruments and other scientific equipment. As universities in the United Kingdom created and expanded physics departments, the governing committee of the observatory became increasingly dominated by paid university physicists in the last two decades of the nineteenth century. By this time, instrument-testing was the observatory's main role. Physicists sought the establishment of a state-funded scientific institution for testing electrical standards.

Founding
The National Physical Laboratory was established in 1900 at Bushy House in Teddington. Its purpose was "for standardising and verifying instruments, for testing materials, and for the determination of physical constants". The laboratory was run by the UK government, with members of staff being part of the civil service. It grew to fill a large selection of buildings on the Teddington site.

Late 20th century
Administration of NPL was contracted out in 1995 under a Government Owned Contractor Operated (GOCO) model, with Serco winning the bid and all staff transferred to their employment. Under this regime, overhead costs halved, third-party revenues grew by 16% per annum, and the number of peer-reviewed research papers published doubled.

NPL procured a large state-of-the-art laboratory under a Private Finance Initiative contract in 1998. The construction was undertaken by John Laing.

21st century
The new laboratory building, which had been maintained by Serco, was transferred back to the DTI in 2004 after the private sector companies involved made losses of over £100m.

It was decided in 2012 to change the operating model for NPL from 2014 onwards to include academic partners and to establish a postgraduate teaching institute on site. The date of the changeover was later postponed for a year. The candidates for lead academic partner were the Universities of Edinburgh, Southampton, Strathclyde and Surrey with an alliance of the Universities of Strathclyde and Surrey chosen as preferred partners.

Funding was announced in January 2013 for a new £25m Advanced Metrology Laboratory that will be built on the footprint of an existing unused building.

The operation of the laboratory transferred back to the Department for Business, Innovation and Skills (now the Department for Science, Innovation and Technology) on 1 January 2015.

Notable researchers
Researchers who have worked at NPL include: D. W. Dye who did important work in developing the technology of quartz clocks; the inventor Sir Barnes Wallis who did early development work on the "Bouncing Bomb" used in the "Dam Busters" wartime raids; H. J. Gough, one of the pioneers of research into metal fatigue, who worked at NPL for 19 years from 1914 to 1938; and Sydney Goldstein and Sir James Lighthill who worked in NPL's aerodynamics division during World War II researching boundary layer theory and supersonic aerodynamics respectively.

Alan Turing, known for his work at the Government Code and Cypher School (GC&CS) at Bletchley Park during the Second World War to decipher German encrypted messages, worked at the National Physical Laboratory from 1945 to 1947. He designed there the ACE (Automatic Computing Engine), which was one of the first designs for a stored-program computer. Clifford Hodge also worked there and was engaged in research on semiconductors. Others who have spent time at NPL include Robert Watson-Watt, generally considered the inventor of radar, Oswald Kubaschewski, the father of computational materials thermodynamics and the numerical analyst James Wilkinson.

Metallurgist Walter Rosenhain appointed the NPL's first female scientific staff members in 1915, Marie Laura Violet Gayler and Isabel Hadfield.

Research
NPL research has contributed to physical science, materials science, computing, and bioscience. Applications have been found in ship design, aircraft development, radar, computer networking, and global positioning.

Atomic clocks
The first accurate atomic clock, a caesium standard based on a certain transition of the caesium-133 atom, was built by Louis Essen and Jack Parry in 1955 at NPL. Calibration of the caesium standard atomic clock was carried out by the use of the astronomical time scale ephemeris time (ET). This led to the internationally agreed definition of the latest SI second being based on atomic time.

Early computers
NPL has undertaken computer research since the mid-1940s. From 1945, Alan Turing led the design of the Automatic Computing Engine (ACE) computer. The ACE project was overambitious and floundered, leading to Turing's departure. Donald Davies took the project over and concentrated on delivering the less ambitious Pilot ACE computer, which first worked in May 1950. Among those who worked on the project was American computer pioneer Harry Huskey. A commercial spin-off, DEUCE was manufactured by English Electric Computers and became one of the best-selling machines of the 1950s.

Packet switching
Beginning in the mid-1960s, Donald Davies invented and pioneered the implementation of packet switching, now the dominant basis for data communications in computer networks worldwide. Davies designed and proposed a national commercial data network in his 1965 Proposal for the Development of a National Communications Service for On-line Data Processing. Subsequently, the NPL team, led by Roger Scantlebury, were the first to implement packet switching in the local-area NPL network in early 1969, which operated until 1986. They carried out work to analyse and simulate the performance of a wide-area packet-switched network capable of providing data communications facilities to most of the U.K. Their research and practice influenced the ARPANET in the United States, the forerunner of the Internet, and other researchers in the UK and Europe, including Louis Pouzin.

NPL sponsors a gallery, opened in 2009, about the development of packet switching and "Technology of the Internet" at The National Museum of Computing.

Internetworking
NPL internetworking research was led by Davies, Barber and Scantlebury, who were members of the International Network Working Group (INWG). Connecting heterogeneous computer networks creates a "basic dilemma" since a common host protocol would require restructuring the existing networks. NPL connected with the European Informatics Network (Barber directed the project and Scantlebury led the UK technical contribution)  by translating between two different host protocols; that is, using a gateway. Concurrently, the NPL connection to the Post Office Experimental Packet Switched Service used a common host protocol in both networks. NPL research confirmed establishing a common host protocol would be more reliable and efficient. The EIN protocol helped to launch the proposed INWG standard. Bob Kahn and Vint Cerf acknowledged Davies and Scantlebury in their 1974 paper "A Protocol for Packet Network Intercommunication".

Scrapbook
Scrapbook was an information storage and retrieval system that went live in mid-1971. It included what would now be called word processing, e-mail and hypertext, anticipating many elements of the World Wide Web. The project was managed by David Yates who said of it "We had a community of bright people that were interested in new things, they were good fodder for a system like Scrapbook" and "When we had more than one Scrapbook system, hyperlinks could go across the network without the user knowing what was happening". It was decided that any commercial development of Scrapbook should be left to industry and it was licensed to Triad and then to BT who marketed it as Milepost and developed a transaction processor as an additional feature. Various implementations were marketed on DEC, IBM and ITL machines. All NPL implementations of Scrapbook were closed down in 1984.

Secure communication
In the early 1990s, the NPL developed three formal specifications of the MAA: one in Z, one in LOTOS, and one in VDM. The VDM specification became part of the 1992 revision of the International Standard 8731–2, and three implementations in C, Miranda, and Modula-2.

Electromagnetics
A 2020 study by researchers from Queen Mary University of London and NPL successfully used microwaves to measure blood-based molecules known to be influenced by dehydration.

Metrology
The National Physical Laboratory is involved with new developments in metrology, such as researching metrology for, and standardising, nanotechnology. It is mainly based at the Teddington site, but also has a site in Huddersfield for dimensional metrology and an underwater acoustics facility at Wraysbury Reservoir near Heathrow Airport.

Directors of NPL
Directors of NPL include a number of notable individuals:
 * Sir Richard Tetley Glazebrook, 1900–1919
 * Sir Joseph Ernest Petavel, 1919–1936
 * Sir Frank Edward Smith, 1936–1937 (acting)
 * Sir William Lawrence Bragg, 1937–1938
 * Sir Charles Galton Darwin, 1938–1949
 * Sir Edward Victor Appleton, 1941 (acting)
 * Sir Edward Crisp Bullard, 1948–1955
 * Reginald Leslie Smith-Rose, 1955–1956 (acting)
 * Sir Gordon Brims Black McIvor Sutherland, 1956–1964
 * John Vernon Dunworth, 1964–1977
 * Paul Dean, 1977–1990
 * Peter Clapham, 1990–1995

Managing Directors
 * John Rae, 1995–2000
 * Bob McGuiness, 2000–2005
 * Steve McQuillan, 2005–2008
 * Martyn Sené, 2008–2009, 2015 (acting)
 * Brian Bowsher, 2009–2015

Chief Executive Officers
 * Peter Thompson, 2015–present