text:
Context-sensitive

Context-sensitive is an adjective meaning "depending on context" or "depending on circumstances". It may refer to:

 * Context-sensitive meaning, where meaning depends on context (language use)
 * Context-sensitive grammar, a formal grammar in which the left-hand sides and right-hand sides of any production rules may be surrounded by a context of terminal and nonterminal symbols
 * Context-sensitive language, a formal language that can be defined by a context-sensitive grammar (and equivalently by a noncontracting grammar). Context-sensitive is one of the four types of grammars in the Chomsky hierarchy
 * Context-sensitive help, a kind of online help that is obtained from a specific point in the state of the software, providing help for the situation that is associated with that state
 * Context-sensitive solutions (also called Context Sensitive Design), a theoretical and practical approach to transportation decision-making and design that takes into consideration the communities and lands through which streets, roads, and highways pass ("the context")
 * Context-sensitive user interface, in computing

title: Context-sensitive


text:
Central America

Central America is a subregion of North America. Its political boundaries are defined as bordering Mexico to the north, Colombia to the south, the Caribbean Sea to the east, and the Pacific Ocean to the west. Central America is usually defined as consisting of seven countries: Belize, Costa Rica, El Salvador, Guatemala, Honduras, Nicaragua, and Panama. Within Central America is the Mesoamerican biodiversity hotspot, which extends from northern Guatemala to central Panama. Due to the presence of several active geologic faults and the Central America Volcanic Arc, there is a high amount of seismic activity in the region, such as volcanic eruptions and earthquakes, which has resulted in death, injury, and property damage.

In the pre-Columbian era, Central America was inhabited by the Indigenous peoples of Mesoamerica to the north and west and the Isthmo-Colombian peoples to the south and east. Following the Spanish expedition of Christopher Columbus' voyages to the Americas, Spain began to colonize the Americas. From 1609 to 1821, the majority of Central American territories (except for what would become Belize and Panama, and including the modern Mexican state of Chiapas) were governed by the viceroyalty of New Spain from Mexico City as the Captaincy General of Guatemala. On 24 August 1821, Spanish Viceroy Juan de O'Donojú signed the Treaty of Córdoba, which established New Spain's independence from Spain. On 15 September 1821, the Act of Independence of Central America was enacted to announce Central America's separation from the Spanish Empire and provide for the establishment of a new Central American state. Some of New Spain's provinces in the Central American region (i.e. what would become Guatemala, Honduras, El Salvador, Nicaragua and Costa Rica) were annexed to the First Mexican Empire; however in 1823 they seceded from Mexico to form the Federal Republic of Central America until 1838.

In 1838, Costa Rica, Guatemala, Honduras, and Nicaragua became the first of Central America's seven states to become independent countries, followed by El Salvador in 1841, Panama in 1903, and Belize in 1981. Despite the dissolution of the Federal Republic of Central America, countries like Costa Rica, El Salvador, Guatemala, Honduras, and Nicaragua continue to maintain a Central American identity. The Belizeans are usually identified as culturally Caribbean rather than Central American, while the Panamanians identify themselves more broadly with their South American neighbors.

The Spanish-speaking countries officially include both North America and South America as a single continent, América, which is split into four subregions: North America (Northern America and Mexico), Central America, South America, and Insular America (the West Indies).

Different definitions

"Central America" may mean different things to various people, based upon different contexts:

 * The non-official United Nations geoscheme for the Americas defines Central America as all states of mainland North America south of the United States, hence grouping Mexico as part of Central America for statistics purposes, but historically and politically Mexico is considered North American.
 * Middle America is usually thought to comprise Mexico to the north of the 7 states of Central America as well as Colombia and Venezuela to the south. Usually, the whole of the Caribbean to the northeast, and sometimes the Guyanas, are also included. According to one source, the term "Central America" was used as a synonym for "Middle America" at least as recently as 1962.
 * In Ibero-America (Spanish and Portuguese speaking American countries), the Americas are considered a single continent (America), and Central America is considered a subregion of the continent comprising the seven countries south of Mexico and north of Colombia.
 * For the people living in the five countries, formerly part of the Federal Republic of Central America there is a distinction between the Spanish language terms "América Central" and "Centroamérica". While both can be translated into English as "Central America", "América Central" is generally used to refer to the geographical area of the seven countries between Mexico and Colombia, while "Centroamérica" is used when referring to the former members of the Federation emphasizing the shared culture and history of the region.
 * In Portuguese as a rule and occasionally in Spanish and other languages, the entirety of the Antilles is often included in the definition of Central America. Indeed, the Dominican Republic is a full member of the Central American Integration System.

History
Central America was formed more than 3 million years ago, as part of the Isthmus of Panama, when its portion of land connected each side of water.

In the Pre-Columbian era, the northern areas of Central America were inhabited by the indigenous peoples of Mesoamerica. Most notable among these were the Mayans, who had built numerous cities throughout the region, and the Aztecs, who had created a vast empire. The pre-Columbian cultures of eastern El Salvador, eastern Honduras, Caribbean Nicaragua, most of Costa Rica and Panama were predominantly speakers of the Chibchan languages at the time of European contact and are considered by some culturally different and grouped in the Isthmo-Colombian Area.

Following the Spanish expedition of Christopher Columbus's voyages to the Americas, the Spanish sent many expeditions to the region, and they began their conquest of Maya territory in 1523. Soon after the conquest of the Aztec Empire, Spanish conquistador Pedro de Alvarado commenced the conquest of northern Central America for the Spanish Empire. Beginning with his arrival in Soconusco in 1523, Alvarado's forces systematically conquered and subjugated most of the major Maya kingdoms, including the K'iche', Tz'utujil, Pipil, and the Kaqchikel. By 1528, the conquest of Guatemala was nearly complete, with only the Petén Basin remaining outside the Spanish sphere of influence. The last independent Maya kingdoms – the Kowoj and the Itza people – were finally defeated in 1697, as part of the Spanish conquest of Petén.

In 1538, Spain established the Real Audiencia of Panama, which had jurisdiction over all land from the Strait of Magellan to the Gulf of Fonseca. This entity was dissolved in 1543, and most of the territory within Central America then fell under the jurisdiction of the Audiencia Real de Guatemala. This area included the current territories of Costa Rica, El Salvador, Guatemala, Honduras, Nicaragua, and the Mexican state of Chiapas, but excluded the lands that would become Belize and Panama. The president of the Audiencia, which had its seat in Antigua Guatemala, was the governor of the entire area. In 1609 the area became a captaincy general and the governor was also granted the title of captain general. The Captaincy General of Guatemala encompassed most of Central America, with the exception of present-day Belize and Panama.

The Captaincy General of Guatemala lasted for more than two centuries, but began to fray after a rebellion in 1811 which began in the Intendancy of San Salvador. The Captaincy General formally ended on 15 September 1821, with the signing of the Act of Independence of Central America. Mexican independence was achieved at virtually the same time with the signing of the Treaty of Córdoba and the Declaration of Independence of the Mexican Empire, and the entire region was finally independent from Spanish authority by 28 September 1821.

From its independence from Spain in 1821 until 1823, the former Captaincy General remained intact as part of the short-lived First Mexican Empire. When the Emperor of Mexico abdicated on 19 March 1823, Central America again became independent. On 1 July 1823, the Congress of Central America peacefully seceded from Mexico and declared absolute independence from all foreign nations, and the region formed the Federal Republic of Central America.

The Federal Republic of Central America was a representative democracy with its capital at Guatemala City. This union consisted of the provinces of Costa Rica, El Salvador, Guatemala, Honduras, Los Altos, Mosquito Coast, and Nicaragua. The lowlands of southwest Chiapas, including Soconusco, initially belonged to the Republic until 1824, when Mexico annexed most of Chiapas and began its claims to Soconusco. The Republic lasted from 1823 to 1838, when it disintegrated as a result of civil wars.

The territory that now makes up Belize was heavily contested in a dispute that continued for decades after Guatemala achieved independence. Spain, and later Guatemala, considered this land a Guatemalan department. In 1862, Britain formally declared it a British colony and named it British Honduras. It became independent as Belize in 1981.

Panama, situated in the southernmost part of Central America on the Isthmus of Panama, has for most of its history been culturally and politically linked to South America. Panama was part of the Province of Tierra Firme from 1510 until 1538 when it came under the jurisdiction of the newly formed Audiencia Real de Panama. Beginning in 1543, Panama was administered as part of the Viceroyalty of Peru, along with all other Spanish possessions in South America. Panama remained as part of the Viceroyalty of Peru until 1739, when it was transferred to the Viceroyalty of New Granada, the capital of which was located at Santa Fé de Bogotá. Panama remained as part of the Viceroyalty of New Granada until the disestablishment of that viceroyalty in 1819. A series of military and political struggles took place from that time until 1822, the result of which produced the republic of Gran Colombia. After the dissolution of Gran Colombia in 1830, Panama became part of a successor state, the Republic of New Granada. From 1855 until 1886, Panama existed as Panama State, first within the Republic of New Granada, then within the Granadine Confederation, and finally within the United States of Colombia. The United States of Colombia was replaced by the Republic of Colombia in 1886. As part of the Republic of Colombia, Panama State was abolished and it became the Isthmus Department. Despite the many political reorganizations, Colombia was still deeply plagued by conflict, which eventually led to the secession of Panama on 3 November 1903. Only after that time did some begin to regard Panama as a North or Central American entity.

By the 1930s the United Fruit Company owned 3.5 e6acre of land in Central America and the Caribbean and was the single largest land owner in Guatemala. Such holdings gave it great power over the governments of small countries. That was one of the factors that led to the coining of the phrase banana republic.

After more than two hundred years of social unrest, violent conflict, and revolution, Central America today remains in a period of political transformation. Poverty, social injustice, and violence are still widespread. Nicaragua is the second poorest country in the western hemisphere (only Haiti is poorer).

Geography

Central America is a part of North America consisting of a tapering isthmus running from the southern extent of Mexico to the northwestern portion of South America. Central America has the Gulf of Mexico, a body of water within the Atlantic Ocean, to the north; the Caribbean Sea, also part of the Atlantic Ocean, to the northeast; and the Pacific Ocean to the southwest. Some physiographists define the Isthmus of Tehuantepec as the northern geographic border of Central America, while others use the northwestern borders of Belize and Guatemala. From there, the Central American land mass extends southeastward to the Atrato River, where it connects to the Pacific Lowlands in northwestern South America.

Central America has over 70 active volcanoes, 41 which are located in El Salvador, and Guatemala. The volcano with the most activity in Central America is Santa María. Still experiencing frequent eruptions to this day, with the last one beginning in 2013, and still is going on to this day.

Of the many mountain ranges within Central America, the longest are the Sierra Madre de Chiapas, the Cordillera Isabelia and the Cordillera de Talamanca. At 4220 m, Volcán Tajumulco is the highest peak in Central America. Other high points of Central America are as listed in the table below:

Between the mountain ranges lie fertile valleys that are suitable for the raising of livestock and for the production of coffee, tobacco, beans and other crops. Most of the population of Honduras, Costa Rica and Guatemala lives in valleys.

Trade winds have a significant effect upon the climate of Central America. Temperatures in Central America are highest just prior to the summer wet season, and are lowest during the winter dry season, when trade winds contribute to a cooler climate. The highest temperatures occur in April, due to higher levels of sunlight, lower cloud cover and a decrease in trade winds.

Central American forests
Central America is part of the Mesoamerican biodiversity hotspot, boasting 7% of the world's biodiversity. The Pacific Flyway is a major north–south flyway for migratory birds in the Americas, extending from Alaska to Tierra del Fuego. Due to the funnel-like shape of its land mass, migratory birds can be seen in very high concentrations in Central America, especially in the spring and autumn. As a bridge between North America and South America, Central America has many species from the Nearctic and the Neotropical realms. However the southern countries (Costa Rica and Panama) of the region have more biodiversity than the northern countries (Guatemala and Belize), meanwhile the central countries (Honduras, Nicaragua and El Salvador) have the least biodiversity. The table below shows recent statistics:

Over 300 species of the region's flora and fauna are threatened, 107 of which are classified as critically endangered. The underlying problems are deforestation, which is estimated by FAO at 1.2% per year in Central America and Mexico combined, fragmentation of rainforests and the fact that 80% of the vegetation in Central America has already been converted to agriculture.

Efforts to protect fauna and flora in the region are made by creating ecoregions and nature reserves. 36% of Belize's land territory falls under some form of official protected status, giving Belize one of the most extensive systems of terrestrial protected areas in the Americas. In addition, 13% of Belize's marine territory are also protected. A large coral reef extends from Mexico to Honduras: the Mesoamerican Barrier Reef System. The Belize Barrier Reef is part of this. The Belize Barrier Reef is home to a large diversity of plants and animals, and is one of the most diverse ecosystems of the world. It is home to 70 hard coral species, 36 soft coral species, 500 species of fish and hundreds of invertebrate species. So far only about 10% of the species in the Belize barrier reef have been discovered.

National trees of Central America
From 2001 to 2010, 5376 km2 of forest were lost in the region. In 2010 Belize had 63% of remaining forest cover, Costa Rica 46%, Panama 45%, Honduras 41%, Guatemala 37%, Nicaragua 29%, and El Salvador 21%. Most of the loss occurred in the moist forest biome, with 12201 km2. Woody vegetation loss was partially set off by a gain in the coniferous forest biome with 4730 km2, and a gain in the dry forest biome at 2054 km2. Mangroves and deserts contributed only 1% to the loss in forest vegetation. The bulk of the deforestation was located at the Caribbean slopes of Nicaragua with a loss of 8574 km2 of forest in the period from 2001 to 2010. The most significant regrowth of 3050 km2 of forest was seen in the coniferous woody vegetation of Honduras.

Montane forests
The Central American pine-oak forests ecoregion, in the tropical and subtropical coniferous forests biome, is found in Central America and southern Mexico. The Central American pine-oak forests occupy an area of 111400 km2, extending along the mountainous spine of Central America, extending from the Sierra Madre de Chiapas in Mexico's Chiapas state through the highlands of Guatemala, El Salvador, and Honduras to central Nicaragua. The pine-oak forests lie between 600 - 1800 m elevation, and are surrounded at lower elevations by tropical moist forests and tropical dry forests. Higher elevations above 1800 m are usually covered with Central American montane forests. The Central American pine-oak forests are composed of many species characteristic of temperate North America including oak, pine, fir, and cypress.

Laurel forest is the most common type of Central American temperate evergreen cloud forest, found in almost all Central American countries, normally more than 1000 m above sea level. Tree species include evergreen oaks, members of the laurel family, species of Weinmannia and Magnolia, and Drimys granadensis. The cloud forest of Sierra de las Minas, Guatemala, is the largest in Central America. In some areas of southeastern Honduras there are cloud forests, the largest located near the border with Nicaragua. In Nicaragua, cloud forests are situated near the border with Honduras, but many were cleared to grow coffee. There are still some temperate evergreen hills in the north. The only cloud forest in the Pacific coastal zone of Central America is on the Mombacho volcano in Nicaragua. In Costa Rica, there are laurel forests in the Cordillera de Tilarán and Volcán Arenal, called Monteverde, also in the Cordillera de Talamanca.

The Central American montane forests are an ecoregion of the tropical and subtropical moist broadleaf forests biome, as defined by the World Wildlife Fund. These forests are of the moist deciduous and the semi-evergreen seasonal subtype of tropical and subtropical moist broadleaf forests and receive high overall rainfall with a warm summer wet season and a cooler winter dry season. Central American montane forests consist of forest patches located at altitudes ranging from 1800 - 4000 m, on the summits and slopes of the highest mountains in Central America ranging from Southern Mexico, through Guatemala, El Salvador, and Honduras, to northern Nicaragua. The entire ecoregion covers an area of 13200 km2 and has a temperate climate with relatively high precipitation levels.

Legendary National Birds of Central America
Ecoregions are not only established to protect the forests themselves but also because they are habitats for an incomparably rich and often endemic fauna. Almost half of the bird population of the Talamancan montane forests in Costa Rica and Panama are endemic to this region. Several birds are listed as threatened, most notably the resplendent quetzal (Pharomacrus mocinno), three-wattled bellbird (Procnias tricarunculata), bare-necked umbrellabird (Cephalopterus glabricollis), and black guan (Chamaepetes unicolor). Many of the amphibians are endemic and depend on the existence of forest. The golden toad that once inhabited a small region in the Monteverde Reserve, which is part of the Talamancan montane forests, has not been seen alive since 1989 and is listed as extinct by IUCN. The exact causes for its extinction are unknown. Global warming may have played a role, because the development of that frog is typical for this area may have been compromised. Seven small mammals are endemic to the Costa Rica-Chiriqui highlands within the Talamancan montane forest region. Jaguars, cougars, spider monkeys, as well as tapirs, and anteaters live in the woods of Central America. The Central American red brocket is a brocket deer found in Central America's tropical forest.

Geology

Central America is geologically very active, with volcanic eruptions and earthquakes occurring frequently, and tsunamis occurring occasionally. Many thousands of people have died as a result of these natural disasters.

Most of Central America rests atop the Caribbean Plate. This tectonic plate converges with the Cocos, Nazca, and North American plates to form the Middle America Trench, a major subduction zone. The Middle America Trench is situated some 60-160 km off the Pacific coast of Central America and runs roughly parallel to it. Many large earthquakes have occurred as a result of seismic activity at the Middle America Trench. For example, subduction of the Cocos Plate beneath the North American Plate at the Middle America Trench is believed to have caused the 1985 Mexico City earthquake that killed as many as 40,000 people. Seismic activity at the Middle America Trench is also responsible for earthquakes in 1902, 1942, 1956, 1982, 1992, January 2001, February 2001, 2007, 2012, 2014, and many other earthquakes throughout Central America.

The Middle America Trench is not the only source of seismic activity in Central America. The Motagua Fault is an onshore continuation of the Cayman Trough which forms part of the tectonic boundary between the North American Plate and the Caribbean Plate. This transform fault cuts right across Guatemala and then continues offshore until it merges with the Middle America Trench along the Pacific coast of Mexico, near Acapulco. Seismic activity at the Motagua Fault has been responsible for earthquakes in 1717, 1773, 1902, 1976, 1980, and 2009.

Another onshore continuation of the Cayman Trough is the Chixoy-Polochic Fault, which runs parallel to, and roughly 80 km to the north, of the Motagua Fault. Though less active than the Motagua Fault, seismic activity at the Chixoy-Polochic Fault is still thought to be capable of producing very large earthquakes, such as the 1816 earthquake of Guatemala.

Managua, the capital of Nicaragua, was devastated by earthquakes in 1931 and 1972.

Volcanic eruptions are also common in Central America. In 1968 the Arenal Volcano, in Costa Rica, erupted killing 87 people as the 3 villages of Tabacon, Pueblo Nuevo and San Luis were buried under pyroclastic flows and debris. Fertile soils from weathered volcanic lava have made it possible to sustain dense populations in the agriculturally productive highland areas.

Life expectancy
List of countries by life expectancy at birth for 2022, according to the World Bank Group. List of Central American countries is expanded by Mexico and Colombia.

Capital cities of Central America
The population of Central America is estimated at $NaN$ as of. With an area of 523780 km2, it has a population density of undefined /km2. Human Development Index values are from the estimates for 2017.

Languages
The official language majority in all Central American countries is Spanish, except in Belize, where the official language is English. Mayan languages constitute a language family consisting of about 26 related languages. Guatemala formally recognized 21 of these in 1996. Xinca, Miskito, and Garifuna are also present in Central America.

Ethnic groups
This region of the continent is very rich in terms of ethnic groups. The majority of the population is mestizo, with sizable Mayan and African descendent populations present, along with numerous other indigenous groups such as the Miskito people. The immigration of Arabs, Jews, Chinese, Europeans and others brought additional groups to the area.

Cathedrals of Central America
The predominant religion in Central America is Christianity (95.6%). Beginning with the Spanish colonization of Central America in the 16th century, Catholicism became the most popular religion in the region until the first half of the 20th century. Since the 1960s, there has been an increase in other Christian groups, particularly Protestantism, as well as other religious organizations, and individuals identifying themselves as having no religion.

Source: Jason Mandrik, Operation World Statistics (2020).

 * Protestantism in Central America also include Independent Christian, most of total Protestants in this region (+80%) are Evangelicals, the rest follow traditional beliefs.
 * Other Christian include Other Traditional Churches (Orthodox, Episcopalian, etc.) and contemporary churches (Mormons, Adventists, Scientology, etc.), also include Non-denominational Christian who are the most numerous group, specially in Guatemala.

National dishes of Central America

 * Central American music
 * Central American cuisine
 * List of cuisines of the Americas – Central American cuisine

Sport

 * Central American Games
 * Central American and Caribbean Games
 * 1926 Central American and Caribbean Games – the first time this event occurred
 * Central American Football Union
 * Surfing

Integration
Central America is currently undergoing a process of political, economic and cultural transformation that started in 1907 with the creation of the Central American Court of Justice.

In 1951 the integration process continued with the signature of the San Salvador Treaty, which created the ODECA, the Organization of Central American States. However, the unity of the ODECA was limited by conflicts between several member states.

In 1991, the integration agenda was further advanced by the creation of the Central American Integration System (Sistema para la Integración Centroamericana, or SICA). SICA provides a clear legal basis to avoid disputes between the member states. SICA membership includes the 7 nations of Central America plus the Dominican Republic, a state that is traditionally considered part of the Caribbean.

On 6 December 2008, SICA announced an agreement to pursue a common currency and common passport for the member nations. No timeline for implementation was discussed.

Central America already has several supranational institutions such as the Central American Parliament, the Central American Bank for Economic Integration and the Central American Common Market.

On 22 July 2011, President Mauricio Funes of El Salvador became the first president pro tempore to SICA. El Salvador also became the headquarters of SICA with the inauguration of a new building.

Foreign relations
Until recently, all Central American countries maintained diplomatic relations with Taiwan instead of China. President Óscar Arias of Costa Rica, however, established diplomatic relations with China in 2007, severing formal diplomatic ties with Taiwan. After breaking off relations with the Republic of China in 2017, Panama established diplomatic relations with the People's Republic of China. In August 2018, El Salvador also severed ties with Taiwan to formally start recognizing the People's Republic of China as sole China, a move many considered lacked transparency due to its abruptness and reports of the Chinese government's desires to invest in the department of La Union while also promising to fund the ruling party's reelection campaign. The President of El Salvador, Nayib Bukele, broke diplomatic relations with Taiwan and established ties with China. On 9 December 2021, Nicaragua resumed relations with the PRC.

Parliament
The Central American Parliament (aka PARLACEN) is a political and parliamentary body of SICA. The parliament started around 1980, and its primary goal was to resolve conflicts in Nicaragua, Guatemala, and El Salvador. Although the group was disbanded in 1986, ideas of unity of Central Americans still remained, so a treaty was signed in 1987 to create the Central American Parliament and other political bodies. Its original members were Guatemala, El Salvador, Nicaragua and Honduras. The parliament is the political organ of Central America, and is part of SICA. New members have since then joined including Panama and the Dominican Republic.

Costa Rica is not a member State of the Central American Parliament and its adhesion remains as a very unpopular topic at all levels of the Costa Rican society due to existing strong political criticism towards the regional parliament, since it is regarded by Costa Ricans as a menace to democratic accountability and effectiveness of integration efforts. Excessively high salaries for its members, legal immunity of jurisdiction from any member State, corruption, lack of a binding nature and effectiveness of the regional parliament's decisions, high operative costs and immediate membership of Central American Presidents once they leave their office and presidential terms, are the most common reasons invoked by Costa Ricans against the Central American Parliament.

Economy

Signed in 2004, the Central American Free Trade Agreement (CAFTA) is an agreement between the United States, Costa Rica, El Salvador, Guatemala, Honduras, Nicaragua, and the Dominican Republic. The treaty is aimed at promoting free trade among its members.

Guatemala has the largest economy in the region. Its main exports are coffee, sugar, bananas, petroleum, clothing, and cardamom. Of its 10.29 billion dollar annual exports, 40.2% go to the United States, 11.1% to neighboring El Salvador, 8% to Honduras, 5.5% to Mexico, 4.7% to Nicaragua, and 4.3% to Costa Rica.

The region is particularly attractive for companies (especially clothing companies) because of its geographical proximity to the United States, very low wages and considerable tax advantages. In addition, the decline in the prices of coffee and other export products and the structural adjustment measures promoted by the international financial institutions have partly ruined agriculture, favouring the emergence of maquiladoras. This sector accounts for 42 per cent of total exports from El Salvador, 55 per cent from Guatemala, and 65 per cent from Honduras. However, its contribution to the economies of these countries is disputed; raw materials are imported, jobs are precarious and low-paid, and tax exemptions weaken public finances.

They are also criticised for the working conditions of employees: insults and physical violence, abusive dismissals (especially of pregnant workers), working hours, non-payment of overtime. According to Lucrecia Bautista, coordinator of the maquilas sector of the audit firm Coverco, "labour law regulations are regularly violated in maquilas and there is no political will to enforce their application. In the case of infringements, the labour inspectorate shows remarkable leniency. It is a question of not discouraging investors." Trade unionists are subject to pressure, and sometimes to kidnapping or murder. In some cases, business leaders have used the services of the maras. Finally, black lists containing the names of trade unionists or political activists are circulating in employers' circles.

Economic growth in Central America is projected to slow slightly in 2014–15, as country-specific domestic factors offset the positive effects from stronger economic activity in the United States.

Central American coast

Tourism in Belize has grown considerably in more recent times, and it is now the second largest industry in the nation. Belizean Prime Minister Dean Barrow has stated his intention to use tourism to combat poverty throughout the country. The growth in tourism has positively affected the agricultural, commercial, and finance industries, as well as the construction industry. The results for Belize's tourism-driven economy have been significant, with the nation welcoming almost one million tourists in a calendar year for the first time in its history in 2012. Belize is also the only country in Central America with English as its official language, making this country a comfortable destination for English-speaking tourists.

Costa Rica is the most visited nation in Central America. Tourism in Costa Rica is one of the fastest growing economic sectors of the country, having become the largest source of foreign revenue by 1995. Since 1999, tourism has earned more foreign exchange than bananas, pineapples and coffee exports combined. The tourism boom began in 1987, with the number of visitors up from 329,000 in 1988, through 1.03 million in 1999, to a historical record of 2.43 million foreign visitors and $1.92-billion in revenue in 2013. In 2012 tourism contributed with 12.5% of the country's GDP and it was responsible for 11.7% of direct and indirect employment.

Tourism in Nicaragua has grown considerably recently, and it is now the second largest industry in the nation. Nicaraguan President Daniel Ortega has stated his intention to use tourism to combat poverty throughout the country. The growth in tourism has positively affected the agricultural, commercial, and finance industries, as well as the construction industry. The results for Nicaragua's tourism-driven economy have been significant, with the nation welcoming one million tourists in a calendar year for the first time in its history in 2010.

Roads
The Inter-American Highway is the Central American section of the Pan-American Highway, and spans 5470 km between Nuevo Laredo, Mexico, and Panama City, Panama. Because of the 87 km break in the highway known as the Darién Gap, it is not possible to cross between Central America and South America in an automobile.

Education

 * List of architecture schools in Central America
 * List of universities in Belize
 * List of universities in Costa Rica
 * List of universities in El Salvador
 * List of universities in Guatemala
 * List of universities in Honduras
 * List of universities in Nicaragua
 * List of universities in Panama

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

title: Central America


text:
Continuous function

In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is. Until the 19th century, mathematicians largely relied on intuitive notions of continuity and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity.

Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the most general continuous functions, and their definition is the basis of topology.

A stronger form of continuity is uniform continuity. In order theory, especially in domain theory, a related concept of continuity is Scott continuity.

As an example, the function $H(t)$ denoting the height of a growing flower at time $t$ would be considered continuous. In contrast, the function $M(t)$ denoting the amount of money in a bank account at time $t$ would be considered discontinuous since it "jumps" at each point in time when money is deposited or withdrawn.

History
A form of the epsilon–delta definition of continuity was first given by Bernard Bolzano in 1817. Augustin-Louis Cauchy defined continuity of $$y = f(x)$$ as follows: an infinitely small increment $$\alpha$$ of the independent variable x always produces an infinitely small change $$f(x+\alpha)-f(x)$$ of the dependent variable y (see e.g. Cours d'Analyse, p. 34). Cauchy defined infinitely small quantities in terms of variable quantities, and his definition of continuity closely parallels the infinitesimal definition used today (see microcontinuity). The formal definition and the distinction between pointwise continuity and uniform continuity were first given by Bolzano in the 1830s, but the work wasn't published until the 1930s. Like Bolzano, Karl Weierstrass denied continuity of a function at a point c unless it was defined at and on both sides of c, but Édouard Goursat allowed the function to be defined only at and on one side of c, and Camille Jordan allowed it even if the function was defined only at c. All three of those nonequivalent definitions of pointwise continuity are still in use. Eduard Heine provided the first published definition of uniform continuity in 1872, but based these ideas on lectures given by Peter Gustav Lejeune Dirichlet in 1854.

Definition

A real function that is a function from real numbers to real numbers can be represented by a graph in the Cartesian plane; such a function is continuous if, roughly speaking, the graph is a single unbroken curve whose domain is the entire real line. A more mathematically rigorous definition is given below.

Continuity of real functions is usually defined in terms of limits. A function $f$ with variable $x$ is continuous at the real number $c$, if the limit of $$f(x),$$ as $x$ tends to $c$, is equal to $$f(c).$$

There are several different definitions of the (global) continuity of a function, which depend on the nature of its domain.

A function is continuous on an open interval if the interval is contained in the function's domain and the function is continuous at every interval point. A function that is continuous on the interval $$(-\infty, +\infty)$$ (the whole real line) is often called simply a continuous function; one also says that such a function is continuous everywhere. For example, all polynomial functions are continuous everywhere.

A function is continuous on a semi-open or a closed interval; if the interval is contained in the domain of the function, the function is continuous at every interior point of the interval, and the value of the function at each endpoint that belongs to the interval is the limit of the values of the function when the variable tends to the endpoint from the interior of the interval. For example, the function $$f(x) = \sqrt{x}$$ is continuous on its whole domain, which is the closed interval $$[0,+\infty).$$

Many commonly encountered functions are partial functions that have a domain formed by all real numbers, except some isolated points. Examples include the reciprocal function $x \mapsto \frac {1}{x}$ and the tangent function $$x\mapsto \tan x.$$ 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.

A partial function is discontinuous at a point if the point belongs to the topological closure of its domain, and either the point does not belong to the domain of the function or the function is not continuous at the point. For example, the functions $x\mapsto \frac {1}{x}$ and $x\mapsto \sin(\frac {1}{x})$  are discontinuous at $0$, and remain discontinuous whichever value is chosen for defining them at $0$. A point where a function is discontinuous is called a discontinuity.

Using mathematical notation, several ways exist to define continuous functions in the three senses mentioned above.

Let $$f : D \to \R$$ be a function defined on a subset $$D$$ of the set $$\R$$ of real numbers.

This subset $$D$$ is the domain of $f$. Some possible choices include
 * $$D = \R $$: i.e., $$ D $$ is the whole set of real numbers. or, for $a$ and $b$ real numbers,
 * $$D = [a, b] = \{x \in \R \mid a \leq x \leq b \} $$: $$ D $$ is a closed interval, or
 * $$D = (a, b) = \{x \in \R \mid a < x < b \} $$: $$ D $$ is an open interval.

In the case of the domain $$D$$ being defined as an open interval, $$a$$ and $$b$$ do not belong to $$D$$, and the values of $$f(a)$$ and $$f(b)$$ do not matter for continuity on $$D$$.

Definition in terms of limits of functions
The function $f$ is continuous at some point $c$ of its domain if the limit of $$f(x),$$ as x approaches c through the domain of f, exists and is equal to $$f(c).$$ In mathematical notation, this is written as $$\lim_{x \to c}{f(x)} = f(c).$$ In detail this means three conditions: first, $f$ has to be defined at $c$ (guaranteed by the requirement that $c$ is in the domain of $f$). Second, the limit of that equation has to exist. Third, the value of this limit must equal $$f(c).$$

(Here, we have assumed that the domain of f does not have any isolated points.)

Definition in terms of neighborhoods
A neighborhood of a point c is a set that contains, at least, all points within some fixed distance of c. Intuitively, a function is continuous at a point c if the range of f over the neighborhood of c shrinks to a single point $$f(c)$$ as the width of the neighborhood around c shrinks to zero. More precisely, a function f is continuous at a point c of its domain if, for any neighborhood $$N_1(f(c))$$ there is a neighborhood $$N_2(c)$$ in its domain such that $$f(x) \in N_1(f(c))$$ whenever $$x\in N_2(c).$$

As neighborhoods are defined in any topological space, this definition of a continuous function applies not only for real functions but also when the domain and the codomain are topological spaces and is thus the most general definition. It follows that a function is automatically continuous at every isolated point of its domain. For example, every real-valued function on the integers is continuous.

Definition in terms of limits of sequences
One can instead require that for any sequence $$(x_n)_{n \in \N}$$ of points in the domain which converges to c, the corresponding sequence $$\left(f(x_n)\right)_{n\in \N}$$ converges to $$f(c).$$ In mathematical notation, $$\forall (x_n)_{n \in \N} \subset D:\lim_{n\to\infty} x_n = c \Rightarrow \lim_{n\to\infty} f(x_n) = f(c)\,.$$

Weierstrass and Jordan definitions (epsilon–delta) of continuous functions
Explicitly including the definition of the limit of a function, we obtain a self-contained definition: Given a function $$f : D \to \mathbb{R}$$ as above and an element $$x_0$$ of the domain $$D$$, $$f$$ is said to be continuous at the point $$x_0$$ when the following holds: For any positive real number $$\varepsilon > 0,$$ however small, there exists some positive real number $$\delta > 0$$ such that for all $$x$$ in the domain of $$f$$ with $$x_0 - \delta < x < x_0 + \delta,$$ the value of $$f(x)$$ satisfies $$f\left(x_0\right) - \varepsilon < f(x) < f(x_0) + \varepsilon.$$

Alternatively written, continuity of $$f : D \to \mathbb{R}$$ at $$x_0 \in D$$ means that for every $$\varepsilon > 0,$$ there exists a $$\delta > 0$$ such that for all $$x \in D$$: $$\left|x - x_0\right| < \delta \text{ implies } |f(x) - f(x_0)| < \varepsilon.$$

More intuitively, we can say that if we want to get all the $$f(x)$$ values to stay in some small neighborhood around $$f\left(x_0\right),$$ we need to choose a small enough neighborhood for the $$x$$ values around $$x_0.$$ If we can do that no matter how small the $$f(x_0)$$ neighborhood is, then $$f$$ is continuous at $$x_0.$$

In modern terms, this is generalized by the definition of continuity of a function with respect to a basis for the topology, here the metric topology.

Weierstrass had required that the interval $$x_0 - \delta < x < x_0 + \delta$$ be entirely within the domain $$D$$, but Jordan removed that restriction.

Definition in terms of control of the remainder
In proofs and numerical analysis, we often need to know how fast limits are converging, or in other words, control of the remainder. We can formalize this to a definition of continuity. A function $$C: [0,\infty) \to [0,\infty]$$ is called a control function if
 * C is non-decreasing
 * $$\inf_{\delta > 0} C(\delta) = 0$$

A function $$f : D \to R$$ is C-continuous at $$x_0$$ if there exists such a neighbourhood $N(x_0)$ that $$|f(x) - f(x_0)| \leq C\left(\left|x - x_0\right|\right) \text{ for all } x \in D \cap N(x_0)$$

A function is continuous in $$x_0$$ if it is C-continuous for some control function C.

This approach leads naturally to refining the notion of continuity by restricting the set of admissible control functions. For a given set of control functions $$\mathcal{C}$$ a function is $\mathcal{C}$-continuous if it is $C$-continuous for some $$C \in \mathcal{C}.$$ For example, the Lipschitz and Hölder continuous functions of exponent $ε$ below are defined by the set of control functions $$\mathcal{C}_{\mathrm{Lipschitz}} = \{C : C(\delta) = K|\delta| ,\ K > 0\}$$ respectively $$\mathcal{C}_{\text{Hölder}-\alpha} = \{C : C(\delta) = K |\delta|^\alpha, \ K > 0\}.$$

Definition using oscillation
Continuity can also be defined in terms of oscillation: a function f is continuous at a point $$x_0$$ if and only if its oscillation at that point is zero; in symbols, $$\omega_f(x_0) = 0.$$ A benefit of this definition is that it discontinuity: the oscillation gives how  the function is discontinuous at a point.

This definition is helpful in descriptive set theory to study the set of discontinuities and continuous points – the continuous points are the intersection of the sets where the oscillation is less than $$\varepsilon$$ (hence a $G_{\delta}$ set) – and gives a rapid proof of one direction of the Lebesgue integrability condition.

The oscillation is equivalent to the $$\varepsilon-\delta$$ definition by a simple re-arrangement and by using a limit (lim sup, lim inf) to define oscillation: if (at a given point) for a given $$\varepsilon_0$$ there is no $$\delta$$ that satisfies the $$\varepsilon-\delta$$ definition, then the oscillation is at least $$\varepsilon_0,$$ and conversely if for every $$\varepsilon$$ there is a desired $$\delta,$$ the oscillation is 0. The oscillation definition can be naturally generalized to maps from a topological space to a metric space.

Definition using the hyperreals
Cauchy defined the continuity of a function in the following intuitive terms: an infinitesimal change in the independent variable corresponds to an infinitesimal change of the dependent variable (see Cours d'analyse, page 34). Non-standard analysis is a way of making this mathematically rigorous. The real line is augmented by adding infinite and infinitesimal numbers to form the hyperreal numbers. In nonstandard analysis, continuity can be defined as follows.

(see microcontinuity). In other words, an infinitesimal increment of the independent variable always produces an infinitesimal change of the dependent variable, giving a modern expression to Augustin-Louis Cauchy's definition of continuity.

Construction of continuous functions

Checking the continuity of a given function can be simplified by checking one of the above defining properties for the building blocks of the given function. It is straightforward to show that the sum of two functions, continuous on some domain, is also continuous on this domain. Given $$f, g \colon D \to \R,$$ then the $$s = f + g$$ (defined by $$s(x) = f(x) + g(x)$$ for all $$x\in D$$) is continuous in $$D.$$

The same holds for the , $$p = f \cdot g$$ (defined by $$p(x) = f(x) \cdot g(x)$$ for all $$x \in D$$) is continuous in $$D.$$

Combining the above preservations of continuity and the continuity of constant functions and of the identity function $$I(x) = x$$ on $\R$, one arrives at the continuity of all polynomial functions on $\R$, such as $$f(x) = x^3 + x^2 - 5 x + 3$$ (pictured on the right).

In the same way, it can be shown that the $$r = 1/f$$ (defined by $$r(x) = 1/f(x)$$ for all $$x \in D$$ such that $$f(x) \neq 0$$) is continuous in $$D\setminus \{x : f(x) = 0\}.$$

This implies that, excluding the roots of $$g,$$ the $$q = f / g$$ (defined by $$q(x) = f(x)/g(x)$$ for all $$x \in D$$, such that $$g(x) \neq 0$$) is also continuous on $$D\setminus \{x:g(x) = 0\}$$.

For example, the function (pictured) $$y(x) = \frac{2x-1}{x+2}$$ is defined for all real numbers $$x \neq -2$$ and is continuous at every such point. Thus, it is a continuous function. The question of continuity at $$x = -2$$ does not arise since $$x = -2$$ is not in the domain of $$y.$$ There is no continuous function $$F : \R \to \R$$ that agrees with $$y(x)$$ for all $$x \neq -2.$$

Since the function sine is continuous on all reals, the sinc function $$G(x) = \sin(x)/x,$$ is defined and continuous for all real $$x \neq 0.$$ However, unlike the previous example, G be extended to a continuous function on  real numbers, by  the value $$G(0)$$ to be 1, which is the limit of $$G(x),$$ when x approaches 0, i.e., $$G(0) = \lim_{x\to 0} \frac{\sin x}{x} = 1.$$

Thus, by setting

G(x) = \begin{cases} \frac {\sin (x)}x & \text{ if }x \ne 0\\ 1 & \text{ if }x = 0, \end{cases} $$

the sinc-function becomes a continuous function on all real numbers. The term is used in such cases when (re)defining values of a function to coincide with the appropriate limits make a function continuous at specific points.

A more involved construction of continuous functions is the function composition. Given two continuous functions $$g : D_g \subseteq \R \to R_g \subseteq \R \quad \text{ and } \quad f : D_f \subseteq \R \to R_f \subseteq D_g,$$ their composition, denoted as $$c = g \circ f : D_f \to \R,$$ and defined by $$c(x) = g(f(x)),$$ is continuous.

This construction allows stating, for example, that $$e^{\sin(\ln x)}$$ is continuous for all $$x > 0.$$

Examples of discontinuous functions
An example of a discontinuous function is the Heaviside step function $$H$$, defined by $$H(x) = \begin{cases} 1 & \text{ if } x \ge 0\\ 0 & \text{ if } x < 0 \end{cases} $$

Pick for instance $$\varepsilon = 1/2$$. Then there is no $\delta$-neighborhood around $$x = 0$$, i.e. no open interval $$(-\delta,\;\delta)$$ with $$\delta > 0,$$ that will force all the $$H(x)$$ values to be within the $\varepsilon$-neighborhood of $$H(0)$$, i.e. within $$(1/2,\;3/2)$$. Intuitively, we can think of this type of discontinuity as a sudden jump in function values.

Similarly, the signum or sign function $$ \sgn(x) = \begin{cases} \;\;\ 1 & \text{ if }x > 0\\ \;\;\ 0 & \text{ if }x = 0\\ -1 & \text{ if }x < 0 \end{cases} $$ is discontinuous at $$x = 0$$ but continuous everywhere else. Yet another example: the function $$f(x) = \begin{cases} \sin\left(x^{-2}\right)&\text{ if }x \neq 0\\ 0&\text{ if }x = 0 \end{cases}$$ is continuous everywhere apart from $$x = 0$$.

Besides plausible continuities and discontinuities like above, there are also functions with a behavior, often coined pathological, for example, Thomae's function, $$f(x)=\begin{cases} 1  &\text{ if } x=0\\ \frac{1}{q}&\text{ if } x = \frac{p}{q} \text{(in lowest terms) is a rational number}\\ 0&\text{ if }x\text{ is irrational}. \end{cases}$$ is continuous at all irrational numbers and discontinuous at all rational numbers. In a similar vein, Dirichlet's function, the indicator function for the set of rational numbers, $$D(x)=\begin{cases} 0&\text{ if }x\text{ is irrational } (\in \R \setminus \Q)\\ 1&\text{ if }x\text{ is rational } (\in \Q) \end{cases}$$ is nowhere continuous.

A useful lemma
Let $$f(x)$$ be a function that is continuous at a point $$x_0,$$ and $$y_0$$ be a value such $$f\left(x_0\right)\neq y_0.$$ Then $$f(x)\neq y_0$$ throughout some neighbourhood of $$x_0.$$

Proof: By the definition of continuity, take $$\varepsilon =\frac{|y_0-f(x_0)|}{2}>0$$, then there exists $$\delta>0$$ such that $$\left|f(x)-f(x_0)\right| < \frac{\left|y_0 - f(x_0)\right|}{2} \quad \text{ whenever } \quad |x-x_0| < \delta$$ Suppose there is a point in the neighbourhood $$|x-x_0|<\delta$$ for which $$f(x)=y_0;$$ then we have the contradiction $$\left|f(x_0)-y_0\right| < \frac{\left|f(x_0) - y_0\right|}{2}.$$

Intermediate value theorem
The intermediate value theorem is an existence theorem, based on the real number property of completeness, and states:

 * If the real-valued function f is continuous on the closed interval $$[a, b],$$ and k is some number between $$f(a)$$ and $$f(b),$$ then there is some number $$c \in [a, b],$$ such that $$f(c) = k.$$

For example, if a child grows from 1 m to 1.5 m between the ages of two and six years, then, at some time between two and six years of age, the child's height must have been 1.25 m.

As a consequence, if f is continuous on $$[a, b]$$ and $$f(a)$$ and $$f(b)$$ differ in sign, then, at some point $$c \in [a, b],$$ $$f(c)$$ must equal zero.

Extreme value theorem
The extreme value theorem states that if a function f is defined on a closed interval $$[a, b]$$ (or any closed and bounded set) and is continuous there, then the function attains its maximum, i.e. there exists $$c \in [a, b]$$ with $$f(c) \geq f(x)$$ for all $$x \in [a, b].$$ The same is true of the minimum of f. These statements are not, in general, true if the function is defined on an open interval $$(a, b)$$ (or any set that is not both closed and bounded), as, for example, the continuous function $$f(x) = \frac{1}{x},$$ defined on the open interval (0,1), does not attain a maximum, being unbounded above.

Relation to differentiability and integrability
Every differentiable function $$f : (a, b) \to \R$$ is continuous, as can be shown. The converse does not hold: for example, the absolute value function
 * $$f(x)=|x| = \begin{cases}

\;\;\ x & \text{ if }x \geq 0\\ -x & \text{ if }x < 0 \end{cases}$$ is everywhere continuous. However, it is not differentiable at $$x = 0$$ (but is so everywhere else). Weierstrass's function is also everywhere continuous but nowhere differentiable.

The derivative f′(x) of a differentiable function f(x) need not be continuous. If f′(x) is continuous, f(x) is said to be continuously differentiable. The set of such functions is denoted $$C^1((a, b)).$$ More generally, the set of functions $$f : \Omega \to \R$$ (from an open interval (or open subset of $$\R$$) $$\Omega$$ to the reals) such that f is $$n$$ times differentiable and such that the $$n$$-th derivative of f is continuous is denoted $$C^n(\Omega).$$ See differentiability class. In the field of computer graphics, properties related (but not identical) to $$C^0, C^1, C^2$$ are sometimes called $$G^0$$ (continuity of position), $$G^1$$ (continuity of tangency), and $$G^2$$ (continuity of curvature); see Smoothness of curves and surfaces.

Every continuous function $$f : [a, b] \to \R$$ is integrable (for example in the sense of the Riemann integral). The converse does not hold, as the (integrable but discontinuous) sign function shows.

Pointwise and uniform limits
Given a sequence $$f_1, f_2, \dotsc : I \to \R$$ of functions such that the limit $$f(x) := \lim_{n \to \infty} f_n(x)$$ exists for all $$x \in D,$$, the resulting function $$f(x)$$ is referred to as the pointwise limit of the sequence of functions $$\left(f_n\right)_{n \in N}.$$ The pointwise limit function need not be continuous, even if all functions $$f_n$$ are continuous, as the animation at the right shows. However, f is continuous if all functions $$f_n$$ are continuous and the sequence converges uniformly, by the uniform convergence theorem. This theorem can be used to show that the exponential functions, logarithms, square root function, and trigonometric functions are continuous.

Directional and semi-continuity
Discontinuous functions may be discontinuous in a restricted way, giving rise to the concept of directional continuity (or right and left continuous functions) and semi-continuity. Roughly speaking, a function is if no jump occurs when the limit point is approached from the right. Formally, f is said to be right-continuous at the point c if the following holds: For any number $$\varepsilon > 0$$ however small, there exists some number $$\delta > 0$$ such that for all x in the domain with $$c < x < c + \delta,$$ the value of $$f(x)$$ will satisfy $$|f(x) - f(c)| < \varepsilon.$$

This is the same condition as continuous functions, except it is required to hold for x strictly larger than c only. Requiring it instead for all x with $$c - \delta < x < c$$ yields the notion of functions. A function is continuous if and only if it is both right-continuous and left-continuous.

A function f is if, roughly, any jumps that might occur only go down, but not up. That is, for any $$\varepsilon > 0,$$ there exists some number $$\delta > 0$$ such that for all x in the domain with $$|x - c| < \delta,$$ the value of $$f(x)$$ satisfies $$f(x) \geq f(c) - \epsilon.$$ The reverse condition is.

Continuous functions between metric spaces
The concept of continuous real-valued functions can be generalized to functions between metric spaces. A metric space is a set $$X$$ equipped with a function (called metric) $$d_X,$$ that can be thought of as a measurement of the distance of any two elements in X. Formally, the metric is a function $$d_X : X \times X \to \R$$ that satisfies a number of requirements, notably the triangle inequality. Given two metric spaces $$\left(X, d_X\right)$$ and $$\left(Y, d_Y\right)$$ and a function $$f : X \to Y$$ then $$f$$ is continuous at the point $$c \in X$$ (with respect to the given metrics) if for any positive real number $$\varepsilon > 0,$$ there exists a positive real number $$\delta > 0$$ such that all $$x \in X$$ satisfying $$d_X(x, c) < \delta$$ will also satisfy $$d_Y(f(x), f(c)) < \varepsilon.$$ As in the case of real functions above, this is equivalent to the condition that for every sequence $$\left(x_n\right)$$ in $$X$$ with limit $$\lim x_n = c,$$ we have $$\lim f\left(x_n\right) = f(c).$$ The latter condition can be weakened as follows: $$f$$ is continuous at the point $$c$$ if and only if for every convergent sequence $$\left(x_n\right)$$ in $$X$$ with limit $$c$$, the sequence $$\left(f\left(x_n\right)\right)$$ is a Cauchy sequence, and $$c$$ is in the domain of $$f$$.

The set of points at which a function between metric spaces is continuous is a $G_{\delta}$ set – this follows from the $$\varepsilon-\delta$$ definition of continuity.

This notion of continuity is applied, for example, in functional analysis. A key statement in this area says that a linear operator $$T : V \to W$$ between normed vector spaces $$V$$ and $$W$$ (which are vector spaces equipped with a compatible norm, denoted $$\|x\|$$) is continuous if and only if it is bounded, that is, there is a constant $$K$$ such that $$\|T(x)\| \leq K \|x\|$$ for all $$x \in V.$$

Uniform, Hölder and Lipschitz continuity
The concept of continuity for functions between metric spaces can be strengthened in various ways by limiting the way $$\delta$$ depends on $$\varepsilon$$ and c in the definition above. Intuitively, a function f as above is uniformly continuous if the $$\delta$$ does not depend on the point c. More precisely, it is required that for every real number $$\varepsilon > 0$$ there exists $$\delta > 0$$ such that for every $$c, b \in X$$ with $$d_X(b, c) < \delta,$$ we have that $$d_Y(f(b), f(c)) < \varepsilon.$$ Thus, any uniformly continuous function is continuous. The converse does not generally hold but holds when the domain space X is compact. Uniformly continuous maps can be defined in the more general situation of uniform spaces.

A function is Hölder continuous with exponent α (a real number) if there is a constant K such that for all $$b, c \in X,$$ the inequality $$d_Y (f(b), f(c)) \leq K \cdot (d_X (b, c))^\alpha$$ holds. Any Hölder continuous function is uniformly continuous. The particular case $$\alpha = 1$$ is referred to as Lipschitz continuity. That is, a function is Lipschitz continuous if there is a constant K such that the inequality $$d_Y (f(b), f(c)) \leq K \cdot d_X (b, c)$$ holds for any $$b, c \in X.$$ The Lipschitz condition occurs, for example, in the Picard–Lindelöf theorem concerning the solutions of ordinary differential equations.

Continuous functions between topological spaces
Another, more abstract, notion of continuity is the continuity of functions between topological spaces in which there generally is no formal notion of distance, as there is in the case of metric spaces. A topological space is a set X together with a topology on X, which is a set of subsets of X satisfying a few requirements with respect to their unions and intersections that generalize the properties of the open balls in metric spaces while still allowing one to talk about the neighborhoods of a given point. The elements of a topology are called open subsets of X (with respect to the topology).

A function $$f : X \to Y$$ between two topological spaces X and Y is continuous if for every open set $$V \subseteq Y,$$ the inverse image $$f^{-1}(V) = \{x \in X \; | \; f(x) \in V \}$$ is an open subset of X. That is, f is a function between the sets X and Y (not on the elements of the topology $$T_X$$), but the continuity of f depends on the topologies used on X and Y.

This is equivalent to the condition that the preimages of the closed sets (which are the complements of the open subsets) in Y are closed in X.

An extreme example: if a set X is given the discrete topology (in which every subset is open), all functions $$f : X \to T$$ to any topological space T are continuous. On the other hand, if X is equipped with the indiscrete topology (in which the only open subsets are the empty set and X) and the space T set is at least T0, then the only continuous functions are the constant functions. Conversely, any function whose codomain is indiscrete is continuous.

Continuity at a point
The translation in the language of neighborhoods of the $(\varepsilon, \delta)$-definition of continuity leads to the following definition of the continuity at a point:

This definition is equivalent to the same statement with neighborhoods restricted to open neighborhoods and can be restated in several ways by using preimages rather than images.

Also, as every set that contains a neighborhood is also a neighborhood, and $$f^{-1}(V)$$ is the largest subset $δ$ of $α$ such that $$f(U) \subseteq V,$$ this definition may be simplified into:

As an open set is a set that is a neighborhood of all its points, a function $$f : X \to Y$$ is continuous at every point of $x$ if and only if it is a continuous function.

If X and Y are metric spaces, it is equivalent to consider the neighborhood system of open balls centered at x and f(x) instead of all neighborhoods. This gives back the above $$\varepsilon-\delta$$ definition of continuity in the context of metric spaces. In general topological spaces, there is no notion of nearness or distance. If, however, the target space is a Hausdorff space, it is still true that f is continuous at a if and only if the limit of f as x approaches a is f(a). At an isolated point, every function is continuous.

Given $$x \in X,$$ a map $$f : X \to Y$$ is continuous at $$x$$ if and only if whenever $$\mathcal{B}$$ is a filter on $$X$$ that converges to $$x$$ in $$X,$$ which is expressed by writing $$\mathcal{B} \to x,$$ then necessarily $$f(\mathcal{B}) \to f(x)$$ in $$Y.$$ If $$\mathcal{N}(x)$$ denotes the neighborhood filter at $$x$$ then $$f : X \to Y$$ is continuous at $$x$$ if and only if $$f(\mathcal{N}(x)) \to f(x)$$ in $$Y.$$ Moreover, this happens if and only if the prefilter $$f(\mathcal{N}(x))$$ is a filter base for the neighborhood filter of $$f(x)$$ in $$Y.$$

Alternative definitions
Several equivalent definitions for a topological structure exist; thus, several equivalent ways exist to define a continuous function.

Sequences and nets
In several contexts, the topology of a space is conveniently specified in terms of limit points. This is often accomplished by specifying when a point is the limit of a sequence. Still, for some spaces that are too large in some sense, one specifies also when a point is the limit of more general sets of points indexed by a directed set, known as nets. A function is (Heine-)continuous only if it takes limits of sequences to limits of sequences. In the former case, preservation of limits is also sufficient; in the latter, a function may preserve all limits of sequences yet still fail to be continuous, and preservation of nets is a necessary and sufficient condition.

In detail, a function $$f : X \to Y$$ is sequentially continuous if whenever a sequence $$\left(x_n\right)$$ in $$X$$ converges to a limit $$x,$$ the sequence $$\left(f\left(x_n\right)\right)$$ converges to $$f(x).$$ Thus, sequentially continuous functions "preserve sequential limits." Every continuous function is sequentially continuous. If $$X$$ is a first-countable space and countable choice holds, then the converse also holds: any function preserving sequential limits is continuous. In particular, if $$X$$ is a metric space, sequential continuity and continuity are equivalent. For non-first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are called sequential spaces.) This motivates the consideration of nets instead of sequences in general topological spaces. Continuous functions preserve the limits of nets, and this property characterizes continuous functions.

For instance, consider the case of real-valued functions of one real variable:

$V$

Proof. Assume that $$f : A \subseteq \R \to \R$$ is continuous at $$x_0$$ (in the sense of $\epsilon-\delta$ continuity). Let $$\left(x_n\right)_{n\geq1}$$ be a sequence converging at $$x_0$$ (such a sequence always exists, for example, $$x_n = x, \text{ for all } n$$); since $$f$$ is continuous at $$x_0$$ $$\forall \epsilon > 0\, \exists \delta_{\epsilon} > 0 : 0 < |x-x_0| < \delta_{\epsilon} \implies |f(x)-f(x_0)| < \epsilon.\quad (*)$$ For any such $$\delta_{\epsilon}$$ we can find a natural number $$\nu_{\epsilon} > 0$$ such that for all $$n > \nu_{\epsilon},$$ $$|x_n-x_0| < \delta_{\epsilon},$$ since $$\left(x_n\right)$$ converges at $$x_0$$; combining this with $$(*)$$ we obtain $$\forall \epsilon > 0 \,\exists \nu_{\epsilon} > 0 : \forall n > \nu_{\epsilon} \quad |f(x_n)-f(x_0)| < \epsilon.$$ Assume on the contrary that $$f$$ is sequentially continuous and proceed by contradiction: suppose $$f$$ is not continuous at $$x_0$$ $$\exists \epsilon > 0 : \forall \delta_{\epsilon} > 0,\,\exists x_{\delta_{\epsilon}}: 0 < |x_{\delta_{\epsilon}}-x_0| < \delta_\epsilon \implies |f(x_{\delta_{\epsilon}})-f(x_0)| > \epsilon$$ then we can take $$\delta_{\epsilon}=1/n,\,\forall n > 0$$ and call the corresponding point $$x_{\delta_{\epsilon}} =: x_n$$: in this way we have defined a sequence $$(x_n)_{n\geq1}$$ such that $$\forall n > 0 \quad |x_n-x_0| < \frac{1}{n},\quad |f(x_n)-f(x_0)| > \epsilon$$ by construction $$x_n \to x_0$$ but $$f(x_n) \not\to f(x_0)$$, which contradicts the hypothesis of sequentially continuity. $$\blacksquare$$

Closure operator and interior operator definitions
In terms of the interior operator, a function $$f : X \to Y$$ between topological spaces is continuous if and only if for every subset $$B \subseteq Y,$$ $$f^{-1}\left(\operatorname{int}_Y B\right) ~\subseteq~ \operatorname{int}_X\left(f^{-1}(B)\right).$$

In terms of the closure operator, $$f : X \to Y$$ is continuous if and only if for every subset $$A \subseteq X,$$ $$f\left(\operatorname{cl}_X A\right) ~\subseteq~ \operatorname{cl}_Y (f(A)).$$ That is to say, given any element $$x \in X$$ that belongs to the closure of a subset $$A \subseteq X,$$ $$f(x)$$ necessarily belongs to the closure of $$f(A)$$ in $$Y.$$ If we declare that a point $$x$$ is a subset $$A \subseteq X$$ if $$x \in \operatorname{cl}_X A,$$ then this terminology allows for a plain English description of continuity: $$f$$ is continuous if and only if for every subset $$A \subseteq X,$$ $$f$$ maps points that are close to $$A$$ to points that are close to $$f(A).$$ Similarly, $$f$$ is continuous at a fixed given point $$x \in X$$ if and only if whenever $$x$$ is close to a subset $$A \subseteq X,$$ then $$f(x)$$ is close to $$f(A).$$

Instead of specifying topological spaces by their open subsets, any topology on $$X$$ can alternatively be determined by a closure operator or by an interior operator. Specifically, the map that sends a subset $$A$$ of a topological space $$X$$ to its topological closure $$\operatorname{cl}_X A$$ satisfies the Kuratowski closure axioms. Conversely, for any closure operator $$A \mapsto \operatorname{cl} A$$ there exists a unique topology $$\tau$$ on $$X$$ (specifically, $$\tau := \{ X \setminus \operatorname{cl} A : A \subseteq X \}$$) such that for every subset $$A \subseteq X,$$ $$\operatorname{cl} A$$ is equal to the topological closure $$\operatorname{cl}_{(X, \tau)} A$$ of $$A$$ in $$(X, \tau).$$ If the sets $$X$$ and $$Y$$ are each associated with closure operators (both denoted by $$\operatorname{cl}$$) then a map $$f : X \to Y$$ is continuous if and only if $$f(\operatorname{cl} A) \subseteq \operatorname{cl} (f(A))$$ for every subset $$A \subseteq X.$$

Similarly, the map that sends a subset $$A$$ of $$X$$ to its topological interior $$\operatorname{int}_X A$$ defines an interior operator. Conversely, any interior operator $$A \mapsto \operatorname{int} A$$ induces a unique topology $$\tau$$ on $$X$$ (specifically, $$\tau := \{ \operatorname{int} A : A \subseteq X \}$$) such that for every $$A \subseteq X,$$ $$\operatorname{int} A$$ is equal to the topological interior $$\operatorname{int}_{(X, \tau)} A$$ of $$A$$ in $$(X, \tau).$$ If the sets $$X$$ and $$Y$$ are each associated with interior operators (both denoted by $$\operatorname{int}$$) then a map $$f : X \to Y$$ is continuous if and only if $$f^{-1}(\operatorname{int} B) \subseteq \operatorname{int}\left(f^{-1}(B)\right)$$ for every subset $$B \subseteq Y.$$

Filters and prefilters
Continuity can also be characterized in terms of filters. A function $$f : X \to Y$$ is continuous if and only if whenever a filter $$\mathcal{B}$$ on $$X$$ converges in $$X$$ to a point $$x \in X,$$ then the prefilter $$f(\mathcal{B})$$ converges in $$Y$$ to $$f(x).$$ This characterization remains true if the word "filter" is replaced by "prefilter."

Properties
If $$f : X \to Y$$ and $$g : Y \to Z$$ are continuous, then so is the composition $$g \circ f : X \to Z.$$ If $$f : X \to Y$$ is continuous and
 * X is compact, then f(X) is compact.
 * X is connected, then f(X) is connected.
 * X is path-connected, then f(X) is path-connected.
 * X is Lindelöf, then f(X) is Lindelöf.
 * X is separable, then f(X) is separable.

The possible topologies on a fixed set X are partially ordered: a topology $$\tau_1$$ is said to be coarser than another topology $$\tau_2$$ (notation: $$\tau_1 \subseteq \tau_2$$) if every open subset with respect to $$\tau_1$$ is also open with respect to $$\tau_2.$$ Then, the identity map $$\operatorname{id}_X : \left(X, \tau_2\right) \to \left(X, \tau_1\right)$$ is continuous if and only if $$\tau_1 \subseteq \tau_2$$ (see also comparison of topologies). More generally, a continuous function $$\left(X, \tau_X\right) \to \left(Y, \tau_Y\right)$$ stays continuous if the topology $$\tau_Y$$ is replaced by a coarser topology and/or $$\tau_X$$ is replaced by a finer topology.

Homeomorphisms
Symmetric to the concept of a continuous map is an open map, for which of open sets are open. If an open map f has an inverse function, that inverse is continuous, and if a continuous map g has an inverse, that inverse is open. Given a bijective function f between two topological spaces, the inverse function $$f^{-1}$$ need not be continuous. A bijective continuous function with a continuous inverse function is called a.

If a continuous bijection has as its domain a compact space and its codomain is Hausdorff, then it is a homeomorphism.

Defining topologies via continuous functions
Given a function $$f : X \to S,$$ where X is a topological space and S is a set (without a specified topology), the final topology on S is defined by letting the open sets of S be those subsets A of S for which $$f^{-1}(A)$$ is open in X. If S has an existing topology, f is continuous with respect to this topology if and only if the existing topology is coarser than the final topology on S. Thus, the final topology is the finest topology on S that makes f continuous. If f is surjective, this topology is canonically identified with the quotient topology under the equivalence relation defined by f.

Dually, for a function f from a set S to a topological space X, the initial topology on S is defined by designating as an open set every subset A of S such that $$A = f^{-1}(U)$$ for some open subset U of X. If S has an existing topology, f is continuous with respect to this topology if and only if the existing topology is finer than the initial topology on S. Thus, the initial topology is the coarsest topology on S that makes f continuous. If f is injective, this topology is canonically identified with the subspace topology of S, viewed as a subset of X.

A topology on a set S is uniquely determined by the class of all continuous functions $$S \to X$$ into all topological spaces X. Dually, a similar idea can be applied to maps $$X \to S.$$

Related notions
If $$f : S \to Y$$ is a continuous function from some subset $$S$$ of a topological space $$X$$ then a of $$f$$ to $$X$$ is any continuous function $$F : X \to Y$$ such that $$F(s) = f(s)$$ for every $$s \in S,$$ which is a condition that often written as $$f = F\big\vert_S.$$ In words, it is any continuous function $$F : X \to Y$$ that restricts to $$f$$ on $$S.$$ This notion is used, for example, in the Tietze extension theorem and the Hahn–Banach theorem. If $$f : S \to Y$$ is not continuous, then it could not possibly have a continuous extension. If $$Y$$ is a Hausdorff space and $$S$$ is a dense subset of $$X$$ then a continuous extension of $$f : S \to Y$$ to $$X,$$ if one exists, will be unique. The Blumberg theorem states that if $$f : \R \to \R$$ is an arbitrary function then there exists a dense subset $$D$$ of $$\R$$ such that the restriction $$f\big\vert_D : D \to \R$$ is continuous; in other words, every function $$\R \to \R$$ can be restricted to some dense subset on which it is continuous.

Various other mathematical domains use the concept of continuity in different but related meanings. For example, in order theory, an order-preserving function $$f : X \to Y$$ between particular types of partially ordered sets $$X$$ and $$Y$$ is continuous if for each directed subset $$A$$ of $$X,$$ we have $$\sup f(A) = f(\sup A).$$ Here $$\,\sup\,$$ is the supremum with respect to the orderings in $$X$$ and $$Y,$$ respectively. This notion of continuity is the same as topological continuity when the partially ordered sets are given the Scott topology.

In category theory, a functor $$F : \mathcal C \to \mathcal D$$ between two categories is called if it commutes with small limits. That is to say, $$\varprojlim_{i \in I} F(C_i) \cong F \left(\varprojlim_{i \in I} C_i \right)$$ for any small (that is, indexed by a set $$I,$$ as opposed to a class) diagram of objects in $$\mathcal C$$.

A is a generalization of metric spaces and posets,  which uses the concept of quantales, and that can be used to unify the notions of metric spaces and domains.

title: Continuous function