Luc Barbet
email: luc.barbet@univ-pau.fr

Unilateral and Bilateral Characterizations of Increasing and Strictly Increasing Maps

Keywords: characterization of isotone (order-preserving, increasing) and strictly isotone maps, left and right isotone maps, totally ordered spaces, complete ordered spaces AMS Classification : 06A05

. Finally, many characterizations of strictly increasing maps are established.

Introduction

The aim of this paper is to give characterizations of isotone (increasing) maps by using some variants of left and right isotone properties. Characterizations of strictly increasing maps are also obtained.

We first recall some classical definitions and notations to avoid ambiguity. If ≤ is a binary relation on E which is reflexive, antisymmetric and transitive, (E, ≤) is an ordered space. An ordered space such that any two elements are comparable is a totally ordered space. An ordered space such that any nonempty majorized subset admits a supremum (and thus, by theorem, any nonempty minorized subset admits an infimum) is a complete ordered space [START_REF] Kelley | General topology[END_REF]. We will write x < y when x ≤ y and x = y. The following notations are defined in [START_REF] Bourbaki | Eléments de mathématiques, Fonctions d'une variable réelle[END_REF]:

[a, →[ = {x ∈ E : a ≤ x} , ]a, →[ = {x ∈ E : a < x} , ]←, b] = {x ∈ E : x ≤ b} , ]←, b[ = {x ∈ E : x < b} , [a, b] = [a, →[ ∩ ]←, b] , ]a, b[ = ]a, →[ ∩ ]←, b[ , [a, b[ = [a, →[ ∩ ]←, b[ , ]a, b] = ]a, →[ ∩ ]←, b] .
These subsets are called intervals, those of the form [a, b] are called segments. Since no confusion is possible when several ordered spaces are considered, we will use the same notations for their order, their intervals, etc.

In the following definitions, f : (A, ≤) → (B, ≤) is a map from an ordered space (A, ≤) into an ordered space (B, ≤). The notion of "preservation of the order" is classical:

Definition 1. f is isotone or increasing if: ∀ (x, y) ∈ A × A, x ≤ y ⇒ f (x) ≤ f (y). (I)
We remark immediately that in this definition (I), formulated from each element of the product space A × A, can be expressed by a right unilateral form, formulated from each element of the subset A + :

(I) ⇔ (I → R ) ⇔ (I → )
where, by definition,

A + := {x ∈ A : ∃y ∈ A, x < y}, (I → R ) ⇔ ∀x ∈ A + , f |[x,→[ is isotone, (I → ) ⇔ ∀x ∈ A + , f ([x, →[) ⊂ [f (x), →[ .
The same is possible with a left unilateral form, formulated from each element of the subset A -:

(I) ⇔ (I ← R ) ⇔ (I ← ) where, by definition,

A -:= {x ∈ A : ∃y ∈ A, y < x}, (I ← R ) ⇔ ∀x ∈ A -, f |]←,x] is isotone, (I ← ) ⇔ ∀x ∈ A -, f (]←, x]) ⊂ ]←, f (x)] .
We now define some various (left or right) lateral variants. We remark that, for left or right lateral versions, saying that f : (A, ≤) → (B, ≤) satisfies the left lateral version means by definition that f : (A, ≥) → (B, ≥) satisfies the right lateral version. Nevertheless we still define the two versions for more completeness.

We first introduce, in an abstract form, a notion defined in [START_REF] Bourbaki | Eléments de mathématiques, Fonctions d'une variable réelle[END_REF] (for a real-valued function defined on an open interval of the real field):

Definition 2. f is left isotone if: ∀x ∈ A -, ∃a ∈ ]←, x[ , f ([a, x]) ⊂ ]←, f (x)] . (I -) f is right isotone if: ∀x ∈ A + , ∃b ∈ ]x, →[ , f ([x, b]) ⊂ [f (x), →[ . (I + )
It is obvious that (I) ⇒ (I -) and (I + ).

(

) 1
The converse implication is false in general (an example is given in § 3). The purpose of § 3 is to determine some sufficient conditions on the ordered spaces so that the isotone property can be obtained by bilateral conditions such as (I -) and (I + ) or by more general conditions that will be defined below.

We first define some notions close to the previous ones by taking into account all possible "directions" (for the left or right side); in other words, we formulate some "star" versions.

Definition 3. f is left star-isotone if: ∀x ∈ A -, ∀c ∈ ]←, x[ , ∃a ∈ [c, x[ , f ([a, x]) ⊂ ]←, f (x)] . (I -) f is right star-isotone if: ∀x ∈ A + , ∀c ∈ ]x, →[ , ∃b ∈ ]x, c] , f ([x, b]) ⊂ [f (x), →[ . (I + )
These notions appear in the theorem of isotone maps under unilateral conditions in § 2. It is easy to prove (I -) ⇒ (I -) and (I + ) ⇒ (I + ); converse implications are false. For instance, (I -) is satisfied but (I -) is not for the map (0, 1) → 1, (1, 0) → 3, (1, 1) → 2 where R 2 is endowed with the product order (the partial order associated with the cone [0, +∞[ 2 ). Nevertheless, when the domain A is totally ordered, we get equivalent properties: (I -) ⇔ (I -) and (I + ) ⇔ (I + ). Proofs are obvious.

Unilateral characterization of isotone maps

We now define some other lateral variants from properties (I → R ) and (I ← R ). Firstly, some "local" versions. Definition 4. f is isotone by left restriction if:

∀x ∈ A -, ∃a ∈ ]←, x[ , f |[a,x] is isotone. (I - R ) f is isotone by right restriction if: ∀x ∈ A + , ∃b ∈ ]x, →[ , f |[x,b] is isotone.
(I + R ) Secondly, the corresponding "star" versions.

Definition 5. f is star-isotone by left restriction if:

∀x ∈ A -, ∀c ∈ ]←, x[ , ∃a ∈ [c, x[ , f |[a,x] is isotone. (I - R ) f is star-isotone by right restriction if: ∀x ∈ A + , ∀c ∈ ]x, →[ , ∃b ∈ ]x, c] , f |[x,b] is isotone. (I + R )
The following table gives some implications between the notions of right lateral increasing that are defined previously.

(I) ⇔ (I → R ) ⇔ (I → ) ⇓ ⇓ (I + R ) ⇒ (I + ) ⇓ ⇓ (I + R ) ⇒ (I + ).
Generally, properties (I + R ), (I + ), (I + R ) or (I + ) are not sufficient to ensure the isotone property. Even if A is countable, totally ordered and complete (and also well ordered); consider for instance

A := {-n -1 : n ∈ N * } ∪ {0}, B := R and f defined by f (0) = 0 and f (-n -1 ) = n (for n ∈ N * ): (I + R
) is satisfied without f being isotone. The purpose of this section is the study of some converse implications relative to the isotone property (I).

We thus introduce the following properties which concern ordered spaces.

Definition 6. An ordered space (A, ≤) is with finite segments if:

∀(x, y) ∈ A × A, x ≤ y ⇒ [x, y] is a finite subset.
If A is a finite subset, (A, ≤) is with finite segments. The converse is false: N and Z are with finite segments. We also remark that this property is not preserved for the closure of a subset of R (consider

I := {n -1 : n ∈ N * } and its closure I ∪ {0}). Definition 7. An ordered space (A, ≤) is with finite connections if: ∀(x, y) ∈ A × A, x ≤ y ⇒ ∃N ∈ N, ∃(a i ) N +1 i=0 ⊂ A, x = a 0 ≤ a i ≤ a i+1 ≤ a N +1 = y and ]a i , a i+1 [ = ∅ for i = 0, 1, ..., N .
If (A, ≤) is with finite segments then it is with finite connections. The converse is false; consider for instance N∪{-∞, +∞} endowed with the order defined by: -∞ (resp. +∞) is the minimum (resp. maximum). When the order is total, (A, ≤) is with finite segments if and only if it is with finite connections; the proof of this result is straightforward.

A unilateral characterization of increasing is given below.

Theorem 1. Let f be a map from an ordered space (A, ≤) into another one (B, ≤). If

A is with finite connections then

(I) ⇔ (I + R ) ⇔ (I + ) ⇔ (I - R ) ⇔ (I -).
Proof. If x ≤ y then, since A is with finite connections, there exists a finite number n of points a i such that x ≤ a 1 ≤ a 2 ≤ ... ≤ a n ≤ y and ]a i , a i+1 [ = ∅ for i = 0, 1, ..., n where a 0 := x and a n+1 := y.

By (I + ), f (a i ) ≤ f (a i+1 ) for i = 0, 1, ..., n since a i+1 is the only point in ]a i , a i+1 ] and thus f ([a i , a i+1 ]) ⊂ [f (a i ), →[. We get f (x) ≤ f (y) by transitivity.
Thus, the first two equivalences are proved. The last two ones can be deduced since (I) is equivalent to the increasing of f from (A, ≥) into (B, ≥).

When the space A is totally ordered, (I + R ) ⇔ (I + R ) and also (I + ) ⇔ (I + ). As a direct consequence, we have the following characterizations.

Corollary 2. If f is a map from a totally ordered space with finite segments (A, ≤) into an ordered space (B, ≤) then

(I) ⇔ (I + R ) ⇔ (I - R ) ⇔ (I + ) ⇔ (I -).
The conclusion of this corollary is not still true under the assumptions of the previous theorem. Indeed, these characterizations are not true if A, even finite, is not totally ordered. Consider for instance the subset A which consists of the points (0, 0), (1, 0) and (0, 1) and is endowed with the order associated with the cone [0, +∞[ 2 and the map f defined by f (0, 0) = 1, f (1, 0) = 2 and f (0, 1) = 0: (I + R ) is satisfied but (I -) is not (and thus f is not isotone).

This corollary cannot be generalized to the case of a lattice (even finite), i.e. an ordered set whose subsets of two elements possess an infimum and a supremum; we can modify the last example by adding the point (1, 1) and by defining f (1, 1) = 2.

Other unilateral characterizations will be established in the following section as a consequence of the main theorem.

Bilateral characterization of isotone maps

In the main theorem which concerns the converse implication of (1) we will use the following lateral notions of increasing that are more general.

Definition 8. f is left quasi-isotone if: ∀x ∈ A -, ∃a ∈ ]←, x[ , f (]a, x]) ⊂ ]←, f (x)] . (QI -) f is right quasi-isotone if: ∀x ∈ A + , ∃b ∈ ]x, →[ , f ([x, b[) ⊂ [f (x), →[ . (QI + )
The converse implications of (I -) ⇒ (QI -) and (I + ) ⇒ (QI + ) are false: any sequence of real numbers is left and right quasi-isotone.

Definition 9. f is pointwise left isotone if: ∀x ∈ A -, ∃a ∈ ]←, x[ , f (a) ∈ ]←, f (x)] . (PI -)
f is pointwise right isotone if:

∀x ∈ A + , ∃b ∈ ]x, →[ , f (b) ∈ [f (x), →[ . (PI + )
Trivially, (I -) ⇒ (P I -) and (I + ) ⇒ (P I + ) but the converse implications are false. For instance, the function defined by f ([-1, 0]) = {0} and f (x) = x -1 if x ∈ ]0, 1] is pointwise right isotone but is not right isotone.

The following notions are also used in the main theorem.

Definition 10. f is left pseudo-isotone or pointwise left star-isotone if:

∀x ∈ A -, ∀c ∈ ]←, x[ , ∃a ∈ [c, x[ , f (a) ∈ ]←, f (x)] . (PI -)
f is right pseudo-isotone or pointwise right star-isotone if:

∀x ∈ A + , ∀c ∈ ]x, →[ , ∃b ∈ ]x, c] , f (b) ∈ [f (x), →[ . (PI + )
It is straightforward that (P I -) ⇒ (P I -) and (P I + ) ⇒ (P I + ) and that the converse implications are false (consider the previous example). When the domain A is well ordered, these implications are not necessarily equivalences; consider the case where A = N: (P I + ) and (P I + ) correspond respectively to the increasing of the sequence and one of its subsequences. We also have the implications (I -) ⇒ (P I -) and (I + ) ⇒ (P I + ).

The following table contents some links between several properties of right lateral increasing; the symbol means the equivalence of the properties when the domain A is totally ordered.

(I) ⇒ (I + R ) ⇒ (I + ) ⇒ (PI + ) ⇓ ⇓ ⇓ (I + R ) ⇒ (I + ) ⇒ (PI + ) ⇓ (QI + ).
We can state the main theorem. Theorem 3. Let (A, ≤) be a complete totally ordered space, (B, ≤) an ordered space and f : (A, ≤) → (B, ≤) a map which is right pseudo-isotone and left quasi-isotone. Then f is isotone.

Proof. Assume that f is not isotone: there exist x 1 < x 2 in A such that the relation f (x 1 ) ≤ f (x 2 ) is not satisfied in the ordered space B. Consider the subset

P := {x ∈ [x 1 , x 2 ] : f (x) f (x 2 )} . ( 2
)
On the one hand, P = ∅ for x 1 ∈ P : f (x 1 ) f (x 2 ). On the other hand, P is a majorized subset: by definition, P ⊂ ]←, x 2 ] . Denote x 0 := sup P (P is a nonempty majorized subset of the complete ordered space A).

We have in particular (x 1 ∈ P and x 2 majorizes P ):

x 1 ≤ x 0 ≤ x 2 . ( 3
)
We also get P ⊂ [x 1 , x 0 ] (for P ⊂ [x 1 , →[ by definition, and x 0 majorizes P ). Now we justify the property:

f (x 0 ) f (x 2 ). (4)
By assuming f (x 0 ) ≤ f (x 2 ), two cases appear. If x 1 = x 0 then f (x 0 ) = f (x 1 ) f (x 2 ) (for x 1 ∈ P ) which contradicts the assumption. Otherwise, x 1 < x 0 and then, since f is left quasi-isotone at x 0 , there exists a < x 0 such that f (]a, x 0 ]) ⊂ ]←, f (x 0 )] and then, from the assumption, f (x) ≤ f (x 2 ) for all x ∈ ]a, x 0 ] . We deduce that P ⊂ ]←, x 0 ] \ ]a, x 0 ] and thus, since A is totally ordered, P ⊂ ]←, a] . Consequently, a majorizes P and thus x 0 ≤ a which gives a contradiction. Thus, property (4) is satisfied. We remark that x 0 ∈ P (by ( 3)-( 4)) and then x 0 = max P. We get:

x 1 ≤ x 0 < x 2 .
(5) Indeed, by [START_REF] Dubreil | [END_REF], it is sufficient to justify that x 0 = x 2 ; this follows from f (x 0 ) = f (x 2 ) (by (4)).

Since f is right pseudo-isotone at x 0 , and x 0 < x 2 (by (5)), there exists

b ∈ ]x 0 , x 2 ] such that f (b) ≥ f (x 0 ). In particular, b ∈ [x 1 , x 2 ] but b / ∈ P (for x 0 < b) which implies that f (b) ≤ f (x 2 ). Thus, f (x 0 ) ≤ f (x 2
) which gives a contradiction with (4).

We can formulate the following bilateral characterizations of the increasing of a map when the order is total and complete on the domain: Obviously, some other equivalences can be formulated above (by changing (QI -) or (P I -) by (I -) and also (QI + ) or (P I + ) by (I + )).

When the domain is N or Z we can simplify the previous characterizations. The set N is discrete (well ordered and such that all elements except the first one has a predecessor [START_REF] Dubreil | [END_REF]) but Z is not; nevertheless Z satisfies the following more general definitions.

Definition 11. An ordered space (Z, ≤) is left quasi-discrete if: ∀x ∈ Z -, ∃a ∈ ]←, x[ , ]a, x[ = ∅; (Z, ≤) is right quasi-discrete if: ∀x ∈ Z + , ∃b ∈ ]x, →[ , ]x, b[ = ∅. (Z, ≤) is quasi-discrete if it is left and right quasi-discrete.
A countable totally ordered space is not necessarily left or right quasi-discrete. The sets

I := {n -1 : n ∈ N * }, J := {-n -1 : n ∈ N * } and I ∪ J are quasi-discrete; I ∪ {0} is left but not right quasi-discrete, J ∪ {0} is right but not left quasi-discrete, I ∪ J ∪ {0} is not left nor right quasi-discrete.
An ordered space with finite connections is quasi-discrete. The converse is false: I ∪ J is quasi-discrete but there exist no finite connections between an element of I and another one of J.

It is immediate that any map from a left (resp. right) quasi-discrete ordered space (Z, ≤) into an ordered space (B, ≤) is left (resp. right) quasi-isotone. We deduce the following corollaries giving unilateral characterizations of the isotone property.

Corollary 5. Let (Z, ≤) be a left quasi-discrete and complete totally ordered space, (B, ≤) an ordered space and f : (Z, ≤) → (B, ≤). Then:

(I) ⇔ (I + R ) ⇔ (I + ) ⇔ (P I + ). In the above corollary, property (P I + ) is not equivalent to the isotone property (consider the map defined on the finite subset {-1, 0, 1} by f (-1) = 0 = f (1) and f (0) = -1). Corollary 6. Let (Z, ≤) be a right quasi-discrete and complete totally ordered space, (B, ≤) an ordered space and f : (Z, ≤) → (B, ≤). Then:

(I) ⇔ (I - R ) ⇔ (I -) ⇔ (P I -
). Property (P I -) is not equivalent to the isotone property even if the domain is finite and well ordered.

The equivalences of one of these two corollaries are not valid under the assumptions of the other one. Consider Z = I ∪ {0} which is a left quasi-discrete and complete totally ordered space and f given by f (0) = 1 and f (i) = i for i ∈ I; this function satisfies (I - R ) (and thus (I -), (P I -)) but is not isotone.

When Z is a quasi-discrete and complete totally ordered space then all properties that appear in the two previous corollaries are equivalent. The fact that the order is total is crucial. Consider Z = 2 N with the inclusion for order; Z is a quasi-discrete and complete partially ordered space. Consider also the function f : Z → {0, 1} given by f (A) = 1 if A is a finite subset of N and f (A) = 0 otherwise. This function is left and right isotone (in fact (I - R ) and (I + R ) hold) but is not isotone.

Strictly isotone maps

The aim of this paragraph is to characterize, by a unilateral or bilateral way, the notion of strictly increasing map (or strictly isotone map), i.e., the property for a map to be isotone relative to the transitive binary relations < associated to the order relations ≤ .

The following definition is classical.

Definition 12. f is strictly increasing or strictly isotone if:

∀ (x, y) ∈ A × A, x < y ⇒ f (x) < f (y). (SI)
We now define the strict versions of various properties of left and right isotone maps. For the notations, we replace formally "I" by "SI".

Definition 13. f is strictly left isotone if:

∀x ∈ A -, ∃a ∈ ]←, x[ , f ([a, x[) ⊂ ]←, f (x)[ . (SI -) f is strictly right isotone if: ∀x ∈ A + , ∃b ∈ ]x, →[ , f (]x, b]) ⊂ ]f (x), →[ . (SI + )
It is obvious that (SI) ⇒ (SI -) and (SI + ).

Definition 14. f is strictly left star-isotone if:

∀x ∈ A -, ∀c ∈ ]←, x[ , ∃a ∈ [c, x[ , f ([a, x[) ⊂ ]←, f (x)[ . (SI -)
f is strictly isotone by left restriction if:

∀x ∈ A -, ∃a ∈ ]←, x[ , f |[a,x] is strictly isotone. (SI - R )
f is strictly star-isotone by left restriction if:

∀x ∈ A -, ∀c ∈ ]←, x[ , ∃a ∈ [c, x[ , f |[a,x] is strictly isotone. (SI - R ) f is left quasi-strict-isotone if: ∀x ∈ A -, ∃a ∈ ]←, x[ , f (]a, x[) ⊂ ]←, f (x)[ . (QSI -) f is pointwise left strict-isotone if: ∀x ∈ A -, ∃a ∈ ]←, x[ , f (a) ∈ ]←, f (x)[ . (PSI -) f is left pseudo-strict-isotone ( or pointwise left star-strict-isotone) if: ∀x ∈ A -, ∀c ∈ ]←, x[ , ∃a ∈ [c, x[ , f (a) ∈ ]←, f (x)[ . (PSI -)
Definition 15. f is strictly right star-isotone if:

∀x ∈ A + , ∀c ∈ ]x, →[ , ∃b ∈ ]x, c] , f (]x, b]) ⊂ ]f (x), →[ . (SI + )
f is strictly isotone by right restriction if:

∀x ∈ A + , ∃b ∈ ]x, →[ , f |[x,b] is strictly isotone. (SI + R )
f is strictly star-isotone by right restriction if:

∀x ∈ A + , ∀c ∈ ]x, →[ , ∃b ∈ ]x, c] , f |[x,b] is strictly isotone. (SI + R ) f is right quasi-strict-isotone if: ∀x ∈ A + , ∃b ∈ ]x, →[ , f (]x, b[) ⊂ ]f (x), →[ . (QSI + ) f is pointwise right strict-isotone if: ∀x ∈ A + , ∃b ∈ ]x, →[ , f (b) ∈ ]f (x), →[ . (PSI + )
f is right pseudo-strict-isotone ( or pointwise right star-strict-isotone) if:

∀x ∈ A + , ∀c ∈ ]x, →[ , ∃b ∈ ]x, c] , f (b) ∈ ]f (x), →[ . (PSI + )
The following table gives some implications between the different properties of left strict-isotone maps. Proof. When x < y, we can consider a finite number of points a i such that a 0 := x < a 1 < a 2 < ... < a n < a n+1 := y and ]a i , a i+1 [ = ∅ for i = 0, 1, ..., n. By (SI + ), f (a i ) < f (a i+1 ) for i = 0, 1, ..., n (since f (]a i , a i+1 ]) ⊂ ]f (a i ), →[ and a i+1 is the only point in ]a i , a i+1 ]).

The first two equivalences are proved and imply the two last ones: (SI) is equivalent to the strict increasing of f from (A, ≥) into (B, ≥).

Corollary 8. If f is a map from a totally ordered space with finite segments (A, ≤) into an ordered space (B, ≤) then (SI) ⇔ (SI + R ) ⇔ (SI - R ) ⇔ (SI + ) ⇔ (SI -).

In the corollary above, the property (P SI + ) is not equivalent to the strict increasing (consider the map defined onto the finite subset {-1, 0, 1} by f (-1) = 0, f (0) = -1 and f (1) = 1).

Corollary 13. Let (Z, ≤) be a right quasi-discrete and complete totally ordered space, (B, ≤) an ordered space and f : (Z, ≤) → (B, ≤). Then:

(SI) ⇔ (SI - R ) ⇔ (SI -) ⇔ (P SI -).

Corollary 4 .

 4 Let (A, ≤) be a complete totally ordered space, (B, ≤) an ordered space and f : (A, ≤) → (B, ≤). Then:(I) ⇔ (I -) and (I + ) ⇔ (QI -) and (P I + ) ⇔ (P I -) and (QI + ).

(

  SI) ⇒ (SI - R ) ⇒ (SI -) ⇒ (PSI -) ⇓ ⇓ ⇓ (SI - R ) ⇒ (SI -) ⇒ (PSI -) ⇓ (QSI -).Some unilateral characterizations of the strict increasing are given bellow.Theorem 7. Let f be a map from an ordered space (A, ≤) into another one (B, ≤). IfA is with finite connections then(SI) ⇔ (SI + R ) ⇔ (SI + ) ⇔ (SI - R ) ⇔ (SI -).

The isotone property and the strictly isotone property can be connected by a general lateral property (P SI + ) or (P SI -).

Proposition 9. Let (A, ≤) and (B, ≤) be ordered spaces and f : (A, ≤) → (B, ≤). If f is isotone and right pseudo-strict-isotone then f is strictly isotone.

Proof. Let x and y in A such that x < y; assumption (I) implies f (x) ≤ f (y). By x ∈ A + and (PSI + ), we can find b ∈ ]x, y] such that f (b) ∈ ]f (x), →[ . We also get f (b) ∈ ]←, f (y)] (for b ≤ y and (I)) and consequently f (x) < f (y), which achieves the proof.

The converse is obvious. Thus, for the general case:

(SI) ⇔ (I) and (P SI + ) ⇔ (I) and (P SI -).

A bilateral condition which ensures the strict increasing can be obtained when the domain is endowed with a complete total order. Theorem 10. Let (A, ≤) be a complete totally ordered space, (B, ≤) an ordered space and f : (A, ≤) → (B, ≤) a map which is right pseudo-strict-isotone and left quasi-isotone. Then f is strictly isotone.

Proof. Since f is right pseudo-isotone (in the strict sense) and left quasi-isotone, f est isotone (by Theorem 3). The last proposition allows to conclude (properties (I) and (PSI + ) imply property (SI)).

We cannot replace the assumption (PSI + ) by the more general condition (PSI + ). Consider A = {0, 1, 2, 3}, B = {0, 1, 2} and the function f which is characterized by its graph {(0, 0), (1, 1), (2, 0), (3, 2)}: this function f is not isotone (and thus is not strictly isotone) but nevertheless is pointwise right strict-isotone and left quasi-isotone.

We obtain the following bilateral characterizations for the strictly isotone property (when the order of the domain of the map is total and complete):

Corollary 11. Let (A, ≤) be a complete totally ordered space, (B, ≤) an ordered space and f : (A, ≤) → (B, ≤). Then:

(SI) ⇔ (SI -) and (I + ) ⇔ (I -) and (SI + ) ⇔ (QI -) and (P SI + ) ⇔ (P SI -) and (QI + ).

Obviously, some other equivalences can be formulated (by replacing (QI -) by (I -) and (QI + ) by (I + ); also by replacing (PSI -) by (SI -) and (PSI + ) by (SI + )).

We remark that the strict assumption on the pseudo-isotone property cannot be associated to the quasi-isotone property: (QSI -) and (P I + ) (similarly (QSI + ) and (P I -)) do not imply the property (SI); consider a constant function onto {0, 1}.

As direct consequences of the previous corollary, some other unilateral characterizations of the strict increasing are immediately obtained.

Corollary 12. Let (Z, ≤) be a left quasi-discrete and complete totally ordered space, (B, ≤) an ordered space and f : (Z, ≤) → (B, ≤). Then:

(SI) ⇔ (SI + R ) ⇔ (SI + ) ⇔ (P SI + ).


Tijani Amahroq

Jean-Paul Penot

Aïcha Syam

On the subdifferentiability of the difference of two functions and local minimization

Keywords: Mathematics Subject Classifications (2000): 26B05, 26B25, 49K27, 90C26 approximately starshaped function, approximately convex function, blunt minimizer, d.a.s. function, d.c. function, equi-subdifferentiability, Fréchet subdifferential, ε-Fréchet subdifferential, gap-continuity, local minimizer, local blunt minimizer

We set up a formula for the Fréchet and ε-Fréchet subdifferentials of the difference of two convex functions. We even extend it to the difference of two approximately starshaped functions. As a consequence of this formula, we give necessary and sufficient conditions for local optimality in nonconvex optimization. Our analysis relies on the notion of gap continuity of multivalued maps and involves concepts of independent interest such as the notions of blunt and sharp minimizers.

1 Introduction.

The interest in the theory of nonsmooth d.c. (difference of convex) optimization problems has much increased in the last few years; see for instance [START_REF] Flores-Bazán | Simplified optimality conditions for minimizing the difference of vector-valued functions[END_REF][START_REF] Hiriart-Urruty | From convex to nonconvex optimization[END_REF][START_REF] Hiriart-Urruty | Generalized differentiability, duality and optimization for problems dealing with differences of convex functions[END_REF][START_REF] Thi | The DC (Difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems[END_REF][START_REF] Michelot | Caractérisation des minima locaux des fonctions de la classe d.c[END_REF][START_REF] Penot | Duality for anticonvex programs[END_REF][START_REF] Penot | On the minimization of difference functions[END_REF][START_REF] An | Convex analysis approach to d. c. programming: Theory, algorithms and applications[END_REF][START_REF] Tuy | On some recent advances and applications of d. c. optimization[END_REF] and references therein. Such an interest is explained by the fact that this class of functions is large: by a result of Hartman [START_REF] Hartman | On functions representable as a difference of convex functions[END_REF], it contains the class of functions of class C 2 ; moreover, the distance function to a nonempty subset of a Hilbert space belongs to this class ( [START_REF] Asplund | Differentiability of the metric projection in finite dimensional Euclidean space[END_REF]). Furthermore, this class contains the class of lower-C 2 functions ( [START_REF] Spingarn | Submonotone subdifferentials of Lipschitz functions[END_REF], [START_REF] Rockafellar | Favorable classes of Lipschitz continuous functions in subgradient optimization[END_REF], [START_REF] Penot | Favorable classes of mappings and multimappings in nonlinear analysis and optimization[END_REF]).

Let f := g -h be defined on a Banach space E with g, h : E → R convex. Michelot [START_REF] Michelot | Caractérisation des minima locaux des fonctions de la classe d.c[END_REF] proved that when E = R n and h is "locally polyhedral" at x, (i.e. ∂h(x) ⊂ ∂h(x) for all x in a neighborhood of x ), then x is a local minimizer of f iff ∂h(x) ⊂ ∂g(x). It is the main aim of the present paper to relax the assumption of [START_REF] Michelot | Caractérisation des minima locaux des fonctions de la classe d.c[END_REF].

Using the Fenchel ε-subdifferential, Hiriart-Urruty [START_REF] Hiriart-Urruty | From convex to nonconvex optimization[END_REF] established that a necessary and sufficient condition for x ∈ E to be an ε-minimizer of f (in the sense that f (x)

≤ inf f (E) + ε) is that ∂ λ h(x) ⊂ ∂ λ+ε g(x)
for all λ ≥ 0.

1 He also proved that x is a global minimizer of f iff

∂ λ h(x) ⊂ ∂ λ g(x)
for all λ ≥ 0.

Considering the functions g(x) = |x| and h(x) = 1 2 x 2 , he showed that this last result is no more valid for local minimizers since 0 is a local minimizer of f := g -h and ∂ λ g(0) = [-1, 1], ∂ λ h(0) = -√ 2λ, √ 2λ , so that the above inclusion is not true for all λ ≥ 0. This is due to the fact that the Fenchel subdifferential is a global notion and cannot give any information about local minimizers.

Our aim in this paper is to give necessary and sufficient conditions for a point x ∈ E to be a local minimizer (or rather a local blunt minimizer in the sense of Definition 6.1 below) in terms of the Fréchet subdifferentials of f or the Fréchet ε-subdifferentials of f . The choice of the Fréchet subdifferential and Fréchet ε-subdifferentials seems to be well suited to our aims; we also deal with other subdifferentials such as the limiting subdifferential. This result will be derived by proving a formula for the Fréchet subdifferential and Fréchet ε-subdifferential of a difference of two approximately starshaped functions or a difference of two (approximately) convex functions.

Preliminaries.

Let f : E → R ∞ := R ∪ {+∞} be a proper extended-real valued function on a Banach space E. We denote by B the closed unit ball of E and by B * the closed unit ball of the dual E * of E.

For ε ≥ 0, the Fenchel ε-subdifferential of f at x ∈ domf := {x ∈ E : f (x) < ∞} is the subset ∂ ε f (x) of the topological dual E * of E defined by ∂ ε f (x) = {x * ∈ E * : f (x) ≥ f (x) + x * , x -x -ε ∀x ∈ E} .
The particular case ε = 0 corresponds to the well known Fenchel subdifferential which is denoted by ∂f (x).

Recall also that the Fréchet ε-subdifferential of f at a point x ∈ domf is defined by

∂ F ε f (x) = x * ∈ E * : lim inf v →0 + f (x + v) -f (x) -x * , v v ≥ -ε (2.1)
and

∂ F ε f (x) = ∅ if x / ∈ domf . More explicitly, for x ∈ domf one has x * ∈ ∂ F ε f (x) ⇔ ∀α > 0 ∃δ > 0 ∀x ∈ B(x, δ) x * , x -x ≤ f (x) -f (x) + (α + ε) x -x . When ε = 0, ∂ F ε f (x) reduces to the well known Fréchet subdifferential of f at x denoted by ∂ F f (x). The Fréchet ε-subdifferential of f at x is clearly related to ∂ F f (x) by ∂ F ε f (x) = ∂ F (f + ε • -x )(x). (2.2) When f is a convex function on E, it is known that ∂ F f (x) = ∂f (x)
, and then

∂ F ε f (x) = ∂f (x) + εB * . (2.3)
If f, g : E → R ∞ are given, as an easy consequence of the definition, for all nonnegative real numbers α, β, one has

∂ F α f (x) + ∂ F β g(x) ⊂ ∂ F α+β (f + g)(x).
(2.4)

The following operation, sometimes called the star-difference, between two subsets C and D of E * will play an important role in the sequel; it is given by

C D := {x * ∈ E * : x * + D ⊂ C}.
It has been used for various purposes (optimal control theory, differential games, morphogenesis, nonsmooth analysis, regularization...) in the literature since its introduction in [START_REF] Pontrjagin | Linear differential games II[END_REF]; see [START_REF] Aubin | Mutational and morphological analysis. Tools for shape evolution and morphogenesis, Systems and Control: Foundations and Applications[END_REF], [START_REF] Caprari | Tangentially d.-s. functions[END_REF], [START_REF] Gautier | Affine and eclipsing multifunctions[END_REF], [START_REF] Martínez-Legaz | Regularization by erasement[END_REF], [START_REF] Matheron | Random sets and integral geometry[END_REF], [START_REF] Penot | Variations on the theme of nonsmooth analysis : another subdifferential[END_REF] and [START_REF] Pshenichnii | Leçons sur les jeux différentiels, in Contrôle optimal et jeux différentiels[END_REF] for some references.

We will use the notion of approximately convex function introduced in [START_REF] Ngai | Approximate convex functions[END_REF] and studied in [START_REF] Aussel | Subsmooth sets: functional characterizations and related concepts[END_REF], [START_REF] Ngai | Approximately convex functions and approximately monotone operators[END_REF] and its generalization presented in [START_REF] Ngai | Semismoothness and directional subconvexity of functions[END_REF]; we recall these notions and present some properties we need.

Definition 2.1 A function f : E → R ∞ := R∪{∞} is
said to be approximately starshaped at x if it is finite at x and if for every ε > 0 one can find δ > 0 such that for every x ∈ B(x, δ) and all t ∈ [0, 1] one has

f ((1 -t)x + tx) ≤ (1 -t)f (x) + tf (x) + εt(1 -t) x -x .
It is said to be approximately convex at x if it is finite at x and if for every ε > 0 one can find δ > 0 such that for every x, x ∈ B(x, δ) and any t ∈ [0, 1] one has

f ((1 -t)x + tx ) ≤ (1 -t)f (x) + tf (x ) + εt(1 -t) x -x .
Both classes enjoy nice stability properties ( [START_REF] Ngai | Approximate convex functions[END_REF], [START_REF] Aussel | Subsmooth sets: functional characterizations and related concepts[END_REF], [START_REF] Ngai | Approximately convex functions and approximately monotone operators[END_REF]). Moreover one has the following result.

Lemma 2.1 (a) If f is approximately starshaped at x, then for any ε > 0 there exists some δ > 0 such that for every

x ∈ B(x, δ), x * ∈ ∂ F f (x) and x * ∈ ∂ F f (x) one has x * , x -x ≤ f (x) -f (x) + ε x -x , (2.5) x * , x -x ≤ f (x) -f (x) + ε x -x .
(2.6) (b) If f is approximately convex at x, then for any ε > 0 there exists some δ > 0 such that for every

x, x ∈ B(x, δ), x * ∈ ∂ F f (x) one has x * , x -x ≤ f (x ) -f (x) + ε x -x .
(2.7)

Proof. (a) Given ε > 0, we choose δ > 0 as in Definition 2.1, so that for every x ∈ B(x, δ), x * ∈ ∂ F f (x) and any t ∈ [0, 1] we have, for some function α : R + → R + satisfying lim t→0 + α(t) = 0,

x * , t(x -x) -tα(t) ≤ f (x + t(x -x)) -f (x) ≤ t(f (x) -f (x)) + εt(1 -t) x -x .
Dividing by t and passing to the limit as t → 0 + , we obtain, for every

x * ∈ ∂ F f (x) x * , x -x ≤ f (x) -f (x) + ε x -x .
Setting s = 1 -t, from the inequality

f (x + s(x -x)) -f (x) ≤ s(f (x) -f (x)) + εs(1 -s) x -x
we deduce similarly the relations

x * , x -x ≤ f (x) -f (x) + ε x -x .
The proof of assertion (b) is similar.

The preceding result shows that approximately convex functions enjoy a very desirable property called firm softness in [START_REF] Penot | Softness, sleekness and regularity properties in nonsmooth analysis[END_REF]. It is related to the concept of limiting subdifferential; recall that the limiting subdifferential ∂ L f (x) of a function f at x is the set of weak * cluster points of bounded sequences

(x * n ) such that x * n ∈ ∂ F f (x n ) for a sequence (x n ) → x satisfying (f (x n )) → f (x) (and then we write (x n ) f → x).
Lemma 2.2 Let f be l.s.c. and approximately convex at x. Then f is firmly soft at x in the following sense: for any sequence (x n ) f → x and any weak * cluster point x * of a bounded sequence

(x * n ) such that x * n ∈ ∂ F f (x n ) for every n ∈ N, one has x * ∈ ∂ F f (x). Equivalently, ∂ L f (x) = ∂ F f (x). Proof. Given sequences (x n ) f → x, x * , (x *
n ) as in the statement, for any ε > 0 there exists some δ > 0 and m ∈ N such that, for every x ∈ B(x, δ), and n ≥ m we have x n ∈ B(x, δ), and

∀x ∈ B(x, δ) x * n , x -x n ≤ f (x ) -f (x n ) + ε x -x n .
Passing to the limit on a subnet, we get, since (

x * n ) is bounded, ∀x ∈ B(x, δ) x * , x -x ≤ f (x ) -f (x) + ε x -x . Thus x * ∈ ∂ F f (x).
A function f is said to be a difference of approximately starshaped functions (in short a d.a.s. function) if there exist two approximately starshaped proper lower semicontinuous (l.s.c.) functions g and h defined on E such that f = g -h. Since g and h may take the value +∞ at the same time, we adopt here the rule (+∞) -(+∞) = +∞. When g and h are convex, f is called a d.c. function.

We will use a mild continuity condition introduced in [START_REF] Penot | Gap continuity of multimaps[END_REF]. We just recall its definition and we refer to [START_REF] Penot | Gap continuity of multimaps[END_REF] for its main properties.

Definition 2.2 A multifunction F : X ⇒ Y between a topological space X and a metric space Y is said to be gap-continuous at

x ∈ X if gap(F (x), F (x)) → 0 as x → x
where, for two subsets A, B of Y one sets

gap(A, B) = inf {d(a, b) : a ∈ A, b ∈ B} , with the usual convention: gap(A, B) = ∞ if A = ∅ or B = ∅.
This notion is weaker than lower and upper semicontinuity. It satisfies some stability properties under composition, sums, products, convex hull operations (see [START_REF] Penot | Gap continuity of multimaps[END_REF]).

3 The Fréchet subdifferentials of d.c. functions.

Let us start our study by recalling a simple but useful observation.

Proposition 3.1 ([38, (7)]) Let g, h : E → R ∞ and let f = g -h be finite at x. Then ∂ F f (x) ⊂ ∂ F g(x) ∂ F h(x). (3.8) In particular, if h is convex, one has ∂ F f (x) ⊂ ∂ F g(x) ∂h(x) and, if g, h are convex, one has ∂ F f (x) ⊂ ∂g(x) ∂h(x).
Proof. When h is finitely valued, the proof follows from the fact that g = f + h, so that

∂ F f (x) + ∂ F h(x) ⊂ ∂ F (f + h)(x) = ∂ F g(x),
but we consider the general case (which could be deduced from the relations f

+ h ≤ g, f (x) + h(x) = g(x)). Given x * ∈ ∂ F f (x), let us show that for any z * ∈ ∂ F h(x) one has y * := x * + z * ∈ ∂ F g(x)
. By definition, for every ε > 0 there exists δ > 0 such that for all x ∈ B(x, δ)

x * , x -x ≤ f (x) -f (x) + ε 2 x -x , z * , x -x ≤ h(x) -h(x) + ε 2 x -x .
The first of these relations shows that f

(x) > -∞ for all x ∈ B(x, δ), so that f (x) + h(x) = g(x).
Adding sides by sides these two inequalities, we get, for all x ∈ B(x, δ),

y * , x -x ≤ g(x) -g(x) + ε x -x , what means that y * ∈ ∂ F g(x).
The last assertion follows from the fact that for a convex function k one has

∂ F k(x) = ∂k(x).
The following immediate consequence can easily be established directly as in [START_REF] Penot | On the minimization of difference functions[END_REF].

Corollary 3.1 If x is a local minimizer of f = g -h, then one has 0 ∈ ∂ F g(x) ∂ F h(x), or equivalently, ∂ F h(x) ⊂ ∂ F g(x).
The following example taken from [START_REF] Penot | On the minimization of difference functions[END_REF] shows that in general the inclusion of Proposition 3.1 is strict. Example 4. Let E = R and let f , g, h be given by f (0) = g(0) = h(0) := 0 and f

(x) = |x| sin 2 (1/x), g(x) = |x|, h(x) = |x| cos 2 (1/x) for x ∈ R\{0}. Then ∂ F g(0) = [-1, 1], ∂ F h(0) = {0}, hence ∂ F g(0) ∂ F h(0) = [-1, 1], while ∂ F f (0) = {0}.
It may happen in particular circumstances that inclusion (3.8) holds as an equality. This is the case for any function f finite at x when one takes g := (r + 1)f, h := rf for some r > 0 (because ∂ F f (x) is convex and for any convex set C one has C = (r + 1)C rC). The following result provides a sufficient condition offering more generality.

Theorem 3.1 Let f = g -h be finite at x, with g, h approximately starshaped at x. Suppose that ∂ F h is gap-continuous at x. Then ∂ F f (x) = ∂ F g(x) ∂ F h(x).
(3.9)

Proof. Taking Proposition 3.1 into account, it suffices to prove that for every

x * ∈ ∂ F g(x) ∂ F h(x) one has x * ∈ ∂ F f (x)
. By definition of gap continuity and Lemma 2.1, given ε > 0 we can find δ > 0 such that

∀x ∈ B(x, δ) ∂ F h(x) (∂ F h(x) + εB * ) = ∅, (3.10) ∀x ∈ B(x, δ), ∀y * ∈ ∂ F g(x) y * , x -x ≤ g(x) -g(x) + ε x -x , (3.11) ∀x ∈ B(x, δ), ∀z * ∈ ∂ F h(x) z * , x -x ≤ h(x) -h(x) + ε x -x . (3.12) Given x ∈ B(x, δ) let us pick z * ∈ ∂ F h(x) and z * ∈ ∂ F h(x) such that z * -z * < ε. Then h(x) is finite and y * := x * + z * ∈ ∂ F g(x)
. Adding sides by sides inequalities (3.11) and (3.12), we get

x * , x -x -z * -z * , x -x ≤ f (x) -f (x) + 2ε x -x , x * , x -x ≤ f (x) -f (x) + 3ε x -x .
Since this last relation is satisfied for every x ∈ B(x, δ), we have

x * ∈ ∂ F f (x).
The special case of d.c. functions follows immediately.

Corollary 3.2 If f := g -h, with g and h convex and if ∂h is gap-continuous at x ∈ domf , then

∂ F f (x) = ∂g(x) ∂h(x).
It is of interest to compare the Fréchet subdifferential ∂ F f (x) of f at x with the limiting subdifferential ∂ L f (x) of f at x when f admits a decomposition g -h.

Theorem 3.2 Let f = g -h, with g, h finite at x, g firmly soft at x and h continuous at x. Suppose that relation (3.9) holds and that ∂ F h is lower semicontinuous at x from X to X * endowed with the weak * topology. Then f is firmly soft at x in the sense that

∂ L f (x) = ∂ F f (x). Proof. Let x * ∈ ∂ L f (x) : there exist sequences (x n ) f → x, (x * n ) bounded in X * , having
x * as a weak * cluster point and satisfying x * n ∈ ∂ F f (x n ) for each n ∈ N. By relation (3.9), it suffices to show that for any z * ∈ ∂ F h(x) we have x * +z * ∈ ∂ F g(x). Let (x n(j) ) j∈J be a subnet of (x n ) n∈N and let (z * j ) j∈J be a net weak * converging to z * such that z * j ∈ ∂h(x n(j) ) for all j ∈ J. Now Proposition 3.

1 ensures that x * n + ∂ F h(x n ) ⊂ ∂ F g(x n ) for each n ∈ N. Thus x * n(j) + z * n(j) ∈ ∂ F g(x n(j)
) for all j ∈ J; passing to the limit, we get x * + z * ∈ ∂ F g(x) as g is firmly soft at x and (g(x n )) → g(x) by the continuity of h at x. Remark. Instead of assuming that (3.9) holds and that g is firmly soft at x one may suppose

∂ F f (x) = ∂ L g(x) ∂ F h(x).
The weak * lower semicontinuity assumption on ∂h is restrictive. However it is satisfied in some cases, as shown by the following corollary. Proof. In cases (a) or (b) ∂ F h is strongly upper semicontinuous at x (see [START_REF] Phelps | Convex Functions, Monotone Operators and Differentiability[END_REF]Lemma 2.6]) and ∂ F h(x) = {Dh(x)}. Moreover, since in case (b) h is continuous around x, in both cases there exists r > 0 such that for any net (x n ) n∈N → x, one can find n 0 ∈ N such that ∂ F h(x n ) is nonempty and contained in rB * for n ≥ n 0 . Then, by weak * compactness of rB * , one can find a subnet (x n(j) ) j∈J of (x n ) n∈N and a weak * converging net (x * j ) j∈J such that x * j ∈ ∂h(x n(j) ) for all j ∈ J. The limit x * of (x * j ) j∈J cannot be anything else that Dh(x).T hus, the lower semicontinuity assumption we used in the preceding theorem is fulfilled. In case (c), setting, for x ∈ X, I(x) := {i ∈ I : h i (x) = h(x)}, we see that I(x) ⊂ I(x) for x close to x and that h i (x) = h j (x) for i, j ∈ I(x) for x close to x, ∂ F h(x) being {h i (x)} for i ∈ I(x). These observations prove that ∂ F h is lower semicontinuous at x; in fact h is of class C 1 at x. Remark. In (a) it would suffice to assume that h is of class T 1 at x in the sense that h is Gâteaux (or Hadamard) differentiable around x and (x, v) → h (x)(v) is continuous at (x, v) for any v ∈ X (see [START_REF] Penot | Favorable classes of mappings and multimappings in nonlinear analysis and optimization[END_REF]).

Corollary 3.3 Let f = g -h,

Equi-subdifferentiability

The following definition delineates a uniformity property which is automatically satisfied when the subdifferential ∂ F f (x) is a singleton, in particular when f is differentiable at x. Thus, it takes into account the multivaluedness of the Fréchet subdifferential.

Definition 4.1 Let us say that a subset

F of ∂ F f (x) is a subset of equi-subdifferentiability of f at x if ∀ε > 0 ∃δ > 0 ∀x * ∈ F ∀x ∈ B(x, δ) f (x) -f (x) -x * , x -x ≥ -ε x -x .
We say that f is equi-subdifferentiable at x if ∂ F f (x) is nonempty and is a subset of equisubdifferentiability of f at x.

Obviously, the convex hull of any finite subset of ∂ F f (x) is a subset of equi-subdifferentiability of f at x. If f is convex, or just approximately starshaped at x, with ∂f (x) nonempty, then f is equi-subdifferentiable at x, as shown by Lemma 2.1 (a) . The same is true if f = g -h with g convex and h Fréchet differentiable at x. In fact, this last assertion is a special case of the next proposition. If g = f + h and if F (resp. H) is a subset of equi-subdifferentiability of f (resp. h) at x, then G := F + H is a subset of equi-subdifferentiability of g at x, as easily seen. If f := sup i∈I f i is the supremum of a family of functions and if, for a finite subset J of I(x) := {i ∈ I : f i (x) = f (x)}, F j is a subset of equi-subdifferentiability of f j at x, then the convex hull of the union of the F j 's is a subset of equi-subdifferentiability of f at x.

If f = g • θ, with θ : X → Y Fréchet differentiable at x and if G is a subset of equi-subdifferentiability of g at θ(x) then F := θ (x) (G) is a subset of equi-subdifferentiability of f at x. Proposition 4.1 Let f = g -h with g, h finite at x and h approximately starshaped at x. If ∂ F h is gap-continuous at x and if G is a subset of equi-subdifferentiability of g at x, then G ∂ F h(x) is a subset of equi-subdifferentiability of f at x. In particular G ∂ F h(x) ⊂ ∂ F f (x).
Proof. The proof is similar to the proof of Theorem 3.1. It suffices to replace ∂ F g(x) by G in relation (3.11).

Taking G := ∂ F g(x) and using Proposition 4.1, we get the following corollary.

Corollary 4.1 If f = g -h with g, h finite at x, g equi-subdifferentiable at x, h approximately starshaped at x and if ∂ F h is gap-continuous at x, then f is equi-subdifferentiable at x and

∂ F f (x) = ∂ F g(x) ∂ F h(x).
Corollary 4.2 If f = g -h with g, h finite at x and approximately starshaped at x and if ∂ F h is gap-continuous at x, then f is equi-subdifferentiable at x and the preceding relation holds.

A converse of the preceding proposition can be given. Its proof follows from an inspection of the proof of Proposition 3.1.

Proposition 4.2 Let f = g -h with g, h finite at x. If F (resp. H) is a subset of equi- subdifferentiability of f (resp. h) at x, then G := F + H is a subset of equi-subdifferentiability of g at x.

Corollary 4.3 Let f = g -h with g, h finite at x, h approximately convex at x, ∂ F h being gap-continuous at x. Then F is a subset of equi-subdifferentiability of f at x, if, and only if,

G := F + ∂ F h(x) is a subset of equi-subdifferentiability of g at x.
Proof. The only if assertion is contained in the preceding proposition. The if assertion stems from Proposition 4.1 and the fact that if

G := F + ∂ F h(x) then F ⊂ G ∂ F h(x).
5 Approximate Fréchet subdifferentials of d.c. functions.

The next theorem exhibits a formula for the ε-Fréchet subdifferential for f := g -h similar to the one in Theorem 3.1. We first note that for every ε ≥ 0 the inclusion

∂ F ε f (x) ⊂ µ≥0 ∂ F ε+µ g(x) ∂ F µ h(x) (5.13)
holds, since we have f + h ≤ g, f (x) + h(x) = g(x), hence, for any µ ≥ 0, by formula (2.4),

∂ F ε f (x) + ∂ F µ h(x) ⊂ ∂ F ε+µ g(x).
In the next theorem we give a sufficient condition ensuring that relation (5.13) is an equality.

Theorem 5.1 Let f := g -h, with g, h approximately starshaped at x. Suppose that, for some λ > 0, the multimap ∂ F λ h is gap-continuous at x. Then, for all ε ≥ 0,

∂ F ε f (x) = ∂ F ε+λ g(x) ∂ F λ h(x) = ∂ F ε+λ g(x) ∂ F λ h(x) = µ≥0 ∂ F ε+µ g(x) ∂ F µ h(x). Proof. Let ε ≥ 0, λ > 0, x * ∈ λ>0 ∂ F ε+λ g(x) ∂ F λ h(x) be given. In view of inclusion (5.13) it suffices to prove that x * ∈ ∂ F ε f (x).
Let us define the functions g ε+λ and h λ by

g ε+λ (x) = g(x) + (ε + λ) x -x , h λ (x) = h(x) + λ x -x .
Then g ε+λ and h λ are approximately starshaped at x and for all x ∈ E one has

g ε+λ (x) -h λ (x) = f ε (x) := f (x) + ε x -x . Theorem 3.1 ensures that x * ∈ ∂ F g ε+λ (x) ∂ F h λ (x) = ∂ F f ε (x) = ∂ F ε f (x). Remark.
In the preceding proof we have used the assumption that the multifunction ∂ F λ h is gap-continuous at x. While for h convex the Fenchel λ-subdifferential ∂ λ h is continuous on domh ( [START_REF] Contesse | Continuity of the approximate subdifferential[END_REF]), the Fréchet λ-subdifferential ∂ F λ h does not seem to be much more regular than ∂h. However, in view of [?, Lemma 3.1, Prop. 3.1 (b)], and of the inclusion

∂ F h(x) + λB * ⊂ ∂ F λ h(x), ∂ F λ h is gap-continuous at x whenever ∂ F h is gap-continuous at x, as easily seen with the help of relation (2.3). Corollary 5.1 Let f = g -h with g, h convex. Suppose ∂h gap-continuous at x. Then, for all ε ≥ 0, ∂ F ε f (x) = (∂g(x) + εB * ) ∂h(x).
6 Application to optimality conditions.

Let us devote this final section to applications of the preceding results to optimality conditions. Our results differ from the ones in [START_REF] Penot | On the minimization of difference functions[END_REF] by several assumptions. Following a number of papers, we will say that a point x ∈ E is a local sharp minimizer (or local firm minimizer) of a function f if f is finite at x and if there exist δ, σ > 0 such that f (x) ≥ f (x) + σ x -x for every x ∈ B(x, δ). Such a property is stronger than mere local strict minimization as it involves a conditioning (or well-posedness) property. On the contrary, the next definition points out weakenings of the notion of minimizer.

Definition 6.1 Let ε be a positive real number. A point x is said to be an ε-minimizer of f on a subset S of E if x ∈ S and if

f (x) -ε ≤ f (x) for all x ∈ S.
We say that x is a blunt minimizer (or, more precisely an ε-blunt minimizer) of f on a subset S of E if x ∈ S and f (x) -ε x -x ≤ f (x) for all x ∈ S.

For S = E we just say that x is a blunt minimizer. We say that x is a local blunt minimizer (or, a local ε-blunt minimizer) of f (on S) if it is an ε-blunt minimizer of f on some neighborhood of x (in S).

An application of Ekeland's variational principle [START_REF] Ekeland | On the variational principle[END_REF] entails part (a) of the following proposition; part (b) is obtained by taking δ = 1. Proposition 6.2 Let f : E → R ∞ be an arbitrary function finite at some x ∈ E. Then x is a local sharp minimizer of f if, and only if, there exists some ρ > 0 such that ρB * is a subset of equi-subdifferentiability of f at x.

Proof. Suppose x is a local sharp minimizer of f : there exist δ > 0 and σ > 0 such that f (x) ≥ f (x) + σ x -x for every x ∈ B(x, δ). Then, for all ρ ∈ (0, σ], all x * ∈ B * and all ε > 0,

x ∈ B(x, δ) we have f (x) -f (x) -x * , x -x + σ x -x ≥ -ε x -x , so that ρB * is a subset of equi-subdifferentiability of f at x.
Conversely, suppose ρB * is a subset of equi-subdifferentiability of f at x for some ρ > 0. Given σ ∈ (0, ρ), let ε > 0 be such that σ + ε ≤ ρ. Let δ > 0 be such that

∀x * ∈ ρB * ∀x ∈ B(x, δ) x * , x -x -ε x -x ≤ f (x) -f (x).
Then, taking the supremum over x * ∈ ρB * , we get

∀x ∈ B(x, δ) σ x -x ≤ (ρ -ε) x -x ≤ f (x) -f (x)
and x is a local sharp minimizer of f .

Corollary 6.1 Let f = g -h with g, h approximately convex at x. If ∂ F h is gap-continuous at x and if 0 ∈ int(∂ F f (x)) (6.14)
then x is a local sharp minimizer of f .

Proof. By assumption, there is some ρ > 0 such that ρB * ⊂ ∂ F f (x). Since ∂ F h is gap-continuous, ρB * is a subset of equi-subdifferentiability of f at x by Corollary 4.2 and the preceding proposition completes the proof.

Remark. The same conclusion holds when g is equi-subdifferentiable at x instead of being approximately convex. Note that assumption (6.14) can also be written: there is some ρ > 0 such that

∂ F h(x) + ρB * ⊂ ∂ F g(x).
We easily recover a result of [START_REF] Penot | On the minimization of difference functions[END_REF].

Corollary 6.2 Let E = R n , f = g -h where g, h are convex and h is continuous at x. If 0 ∈ int(∂ F f (x)), then x is a local sharp minimizer of f .
Proof. This is a direct consequence of the fact that ∂h is gap continuous at xwhen E is finite dimensional and h is continuous at x by [START_REF] Penot | Gap continuity of multimaps[END_REF] and of Corollary 6.1.

The following result gives necessary and sufficient conditions for local blunt optimality for d.c. or d.a.s. functions. Theorem 6.1 Suppose f := g -h with g, h approximately starshaped and finite at x and that ∂ F h is gap-continuous at x. Then the following assertions are equivalent:

i) ∂ F h(x) ⊂ ∂ F g(x); ii) 0 ∈ ∂ F f (x); iii) for all ε > 0, x is a local ε-blunt minimizer of f ; iv) for some λ > 0, ∂ F λ h(x) ⊂ ∂ F λ g(x); v) for all λ > 0, ∂ F λ h(x) ⊂ ∂ F λ g(x).
Proof. The equivalence ii)⇔iii) is immediate and valid for any function f . The equivalence i)⇔ii) is a consequence of Theorem 3.1. The equivalence ii)⇔iv) stems from Theorem 5.1 with ε = 0 since, as observed in the remark following Theorem 5.1, ∂ F λ h is gap-continuous whenever ∂ F h is gap-continuous. The equivalence ii)⇔ v) is also a consequence of Theorem 5.1 Example 5. For the example considered in the introduction, f = g -h with g(x) = |x| and h(x) = 1 2 x 2 , 0 is a local ε-blunt minimizer of f for all ε > 0 (in fact it is a local minimizer) and we also have for all λ > 0 ∂

F λ h(0) ⊂ ∂ F λ g(0) since a direct calculation shows that ∂ F λ g(0) = [-1 -λ, 1 + λ] and ∂ F λ h(0) = [-λ, λ] .
For f = g -h with g = 0, h(x) = 1 2 x 2 , 0 is a local ε-blunt minimizer of f for all ε > 0 as condition i) above is satisfied, but 0 is not a local minimizer of f . Now let us consider a general constrained minimization problem of the form:

(P C ) minimize f (x) := g(x) -h(x) subject to x ∈ C

where g, h : E → R are specified below and C is a nonempty closed convex subset of E. Proposition 6.3 Assume f := g -h with g approximately convex and continuous at x. Then a necessary condition for x ∈ C to be a local ε-blunt minimizer of f on C for all ε > 0 is that

∂ F h(x) ⊂ ∂ F g(x) + N C (x)
where N C (x) is the normal cone of convex analysis. Moreover, if h is approximately starshaped at x and ∂ F h is gap-continuous at x, then the above inclusion is a sufficient condition for x to be a local ε-blunt minimizer of f on C for all ε > 0.

∂ F h(x) ⊂ N C (x).
Moreover if h is approximately starshaped at x and ∂ F h is gap-continuous at x, then the above inclusion is a sufficient condition for x to be an ε-blunt maximizer of h -ι C , i.e. an ε-blunt maximizer of h on C.

Proposition 6 . 1

 61 Let C be a nonempty closed subset of E and let f : E → R ∞ be a l.s.c. function. (a) If x 0 is an ε-minimizer of f on C then, for all δ > 0, there exists x ∈ C ∩ B(x 0 , δ) such that x is an ε δ -blunt minimizer of f on C. (b) If f is bounded below on C then, for all ε > 0, there exists an ε-blunt minimizer of f on C.

Proof.

  If for all ε > 0, x ∈ C is a local ε-blunt minimizer of f on C, then, for all ε > 0, x is a local ε-bluntminimizer of f + ι C on E where ι C is the indicator function of C. Therefore 0 ∈ ∂ F (f +ι C )(x). Applying Proposition 3.1, one gets ∂ F h(x) ⊂ ∂ F (g +ι C )(x) = ∂ F g(x)+ N C (x), by the usual sum rule for limiting subdifferentials ([START_REF] Fabian | Subdifferentials, local ε -supports and Asplund spaces[END_REF]) since g is locally Lipschitzian around x by[START_REF] Ngai | Approximate convex functions[END_REF] and since g is firmly soft at x by Lemma 2.2.Conversely, if ∂ F h is gap-continuous and if the relation∂ F h(x) ⊂ ∂ F g(x) + N C (x) is satisfied, Theorem 3.1 ensures that 0 ∈ ∂ F (f + ι C )(x) since f + ι C = (g + ι C ) -h and ∂ F (g + ι C )(x) = ∂ F g(x) + N C (x). By Theorem 6.1 x is an ε-blunt minimizer of f + ι C and then x is an ε-blunt minimizer of f on C. Now we are interested in the following maximization problems:(M C ) maximize h(x) subject to x ∈ C where h : E → R is a function and C is a nonempty closed convex subset of E. It is clear that sup x∈C h(x) = -inf x∈E {ι C (x) -h(x)}.So, a necessary condition for x to be an ε-blunt maximizer of h on C (i.e. x is an ε-blunt minimizer of -h on C) is

  with g, h finite at x, g firmly soft at x. Suppose relation (3.9) holds. Then, under each of the following assumptions, f is firmly soft at x :

(a)

h is of class C 1 at x (i.e. h is differentiable around x and its derivative is continuous at x);

(b) h is convex and Fréchet differentiable at x; (c) h is the infimum of a finite family (h i ) i∈I of functions of class C 1 at x and ∂ F h(x) is nonempty for x in a neighborhood of x.

Acknowledgements. The authors are most grateful to C. Zȃlinescu for his helpful remarks on an earlier version of the manuscript.


Jean-Paul Penot

Delineating nice classes of nonsmooth functions

Keywords: Mathematics Subject Classification: 49J52, 46N10, 46T20 approximately convex function, equi-subdifferentiability, monotonicity, nonsmooth analysis, normal, semi-monotonicity, subdifferential, uniform subdifferentiability

published or not. The documents may come

Delineating nice classes of nonsmooth functions

Introduction

While it is known thanks to Weierstrass that wild nonsmooth functions exist, it is of interest to select classes of nonsmooth functions which have pleasant properties. Such an appraisal can be understood in different manners: [START_REF] Amahroq | Subdifferentiation and minimization of the difference of two functions[END_REF] one may hope to get properties close to the ones of convex or differentiable functions, (2) one may expect stability properties with respect to the usual operations, (3) one may wish that the main concepts of nonsmooth analysis coincide for such classes, (4) one may devote efforts to show that the regularization of such functions is worthwhile.

Many works have been devoted to such aims. Central concepts are the notions of approximately convex functions ( [START_REF] Aussel | Subsmooth sets: functional characterizations and related concepts[END_REF], [START_REF] Ngai | Approximate convex functions[END_REF], [START_REF] Ngai | Approximately convex functions and approximately monotone operators[END_REF]) and paraconvex (or semiconvex) functions ( [START_REF] Canino | On p-convex sets and geodesics[END_REF], [START_REF] Cannarsa | Semiconcave Functions, Hamilton-Jacobi Equations and Optimal Control[END_REF], [START_REF] Ngai | Paraconvex functions and paraconvex sets[END_REF], [START_REF] Rolewicz | On the coincidence of some subdifferentials in the class of α(•)-paraconvex functions[END_REF]- [START_REF] Rolewicz | Paraconvex analysis[END_REF], [START_REF] Vial | Strong and weak convexity of sets and functions[END_REF]...). They can be given several variants. Directional variants have been studied in [START_REF] Daniilidis | Approximate convexity and submonotonicity[END_REF], [START_REF] Daniilidis | Integration of multivalued operators and cyclic submonotonicity[END_REF], in particular in connection with integration questions. Another line of research appears in the works of Poliquin, Rockafellar, Thibault and their co-authors ( [START_REF] Bernard | Prox-regularity of functions and sets in Banach spaces[END_REF], [START_REF] Bernard | Prox-regular functions in Hilbert spaces[END_REF], [START_REF] Jourani | Weak regularity of functions and sets in Asplund spaces[END_REF], [START_REF] Poliquin | Integration of subdifferentials of nonconvex functions[END_REF], [START_REF] Poliquin | Prox-regular functions in variational analysis[END_REF], [START_REF] Poliquin | Local differentiability of distance functions[END_REF]...) . In such works, the subjet of the function is heavily present, even if it is not put to the fore. It is the purpose of the present paper to put it in clear light and to propose some variants. We keep in mind expectations (1)-(3), in particular [START_REF] Aubin | Set-Valued Analysis[END_REF] and what has been called softness in [START_REF] Penot | Softness, sleekness and regularity in nonsmooth analysis[END_REF] and lower regularity in [START_REF] Mordukhovich | Variational analysis and generalized differentiation[END_REF]. We refer to [START_REF] Attouch | Approximation and regularization of arbitrary functions in Hilbert spaces by the Lasry-Lions method[END_REF], [START_REF] Bougeard | Towards minimal assumptions for the infimal convolution regularization[END_REF], [START_REF] Martínez-Legaz | Regularization by erasement[END_REF], [START_REF] Ngai | ϕ-regular functions in Asplund spaces[END_REF], [START_REF] Rockafellar | Variational Analysis[END_REF] and their references for the question of regularization. With [START_REF] Jourani | Radiality and semismoothness[END_REF], [START_REF] Jourani | Weak regularity of functions and sets in Asplund spaces[END_REF] and [START_REF] Ngai | Semismoothness and directional subconvexity of functions[END_REF], the paper [START_REF] Ngai | ϕ-regular functions in Asplund spaces[END_REF] has been our starting point.

Our main thrusts consist in putting to the fore different localizations in the subjet of the function and in clarifying the roles of different notions of uniform or equi-differentiability. Because the passages from functions to sets and from sets to functions have been dealt with in several other papers ( [START_REF] Bounkel | On various notions of regularity of sets in nonsmooth analysis[END_REF], [START_REF] Borwein | Techniques of Variational Analysis[END_REF], [START_REF] Colombo | Variational inequalities and regularity properties of closed sets in Hilbert spaces[END_REF], [START_REF] Jourani | Weak regularity of functions and sets in Asplund spaces[END_REF], [START_REF] Ngai | Approximately convex functions and approximately monotone operators[END_REF]- [START_REF] Ngai | ϕ-regular functions in Asplund spaces[END_REF], [START_REF] Rockafellar | Variational Analysis[END_REF]...), for the sake of brevity, we do not explore all the possibilities but just present some of them.

Preliminaries

In the sequel X is a Banach space with dual space X * and F(X) (resp. S(X)) denotes the set of all functions (resp. lower semicontinuous (l.s.c.) functions) from X to R ∞ := R ∪ {+∞}. For more information about the basic concepts of nonsmooth analysis we recall in this section, we refer to the recent monographs of Borwein and Zhu ( [START_REF] Borwein | Techniques of Variational Analysis[END_REF]), Mordukhovich [START_REF] Mordukhovich | Variational analysis and generalized differentiation[END_REF], Rockafellar and Wets [START_REF] Rockafellar | Variational Analysis[END_REF], Schirotzek ([49]) and Zalinescu [START_REF] Zalinescu | Convex Analysis in General Vector Spaces[END_REF].

The lower directional derivative (or contingent derivative or Dini-Hadamard derivative) of f ∈ F(X) finite at x is given by

f D (x, v) := lim inf (t,w)→(0 + ,v) 1 t (f (x + tw) -f (x)).
The Clarke-Rockafellar derivative [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF], or circa-derivative of f ∈ S(X) at x is given by

f C (x, v) := sup r>0 lim sup (t,y)→(0 + ,x) f (y)→f (x) inf w∈B(v,r) 1 t (f (y + tw) -f (y)).
The directional (or Dini-Hadamard) subdifferential and the circa (or Clarke-Rockafellar) subdifferential are associated to these derivatives by

∂ D f (x) = {x * ∈ X * : x * ≤ f D (x, •)}, ∂ C f (x) = {x * ∈ X * : x * ≤ f C (x, •)}.
The firm (or Fréchet) subdifferential of f ∈ F (X) at x is the set ∂ F f (x) of x * ∈ X * such that for all ε > 0 there exists some δ > 0 for which

f (w) -f (x) -x * , w -x ≥ -ε w -x ∀w ∈ B(x, δ).
Given an hyper-modulus π, i.e. a function π : R

+ → R + ∪ {+∞} such that lim r→0+ π(r)/r = 0, π(0) = 0, ∂ F f (x) contains the π-proximal subdifferential of f at x which is the set ∂ π f (x) of x * ∈ X * such that f (w) -f (x) -x * , w -x ≥ -π( w -x ) ∀w ∈ X. For π(t) := t 2 , ∂ π f (x) is just called the proximal subdifferential of f at x. In fact, ∂ F f (x)
is the union of the π-proximal subdifferentials of f at x for π in the set of hyper-modulus.

The limiting subdifferential of f at x is defined as

∂ L f (x) := lim sup y→ f x ∂ F f (y)
, where y → f x means that y → x with f (y) → f (x), the limsup being taken in a sequential way with respect to the weak * topology.

As in [START_REF] Penot | Softness, sleekness and regularity in nonsmooth analysis[END_REF], we say that f is soft at x if the limiting subdifferential

∂ L f (x) of f at x coincides with ∂ F f (x).
We call subdifferential a multifunction ∂ : F(X) × X ⇒ X * such that for all f ∈ F(X) and all x ∈ X one has ∂ F f (x) ⊂ ∂f (x) ⊂ ∂ C f (x). These inclusions imply that for all f ∈ F(X) the domain of ∂f is included in the domain of f and that ∂f is locally bounded if f is locally Lipschitzian. The inclusion ∂ F f ⊂ ∂f could be relaxed into ∂ π f ⊂ ∂f for some hyper-modulus π, but we require it for the sake of simplicity.

Given a subdifferential ∂, recall that the ∂-subjet (in short subjet) of f ∈ F(X) is the set

J ∂ f := {(x, x * , f (x)) : x ∈ X, f (x) < +∞, x * ∈ ∂f (x)}.
This notion generalizes the concept of one-jet classically used in differential geometry. It will play a crucial role in [START_REF] Ioffe | Limiting subhessians, limiting subjets and their calculus[END_REF], [START_REF] Penot | Critical duality[END_REF], [START_REF] Penot | Ekeland duality as a paradigm[END_REF] and in what follows.

A spectrum of subdifferentiable functions

In the sequel, f ∈ S(X), i.e. f : X → R ∞ is a l.s.c. function and x ∈ dom f := {x ∈ X : f (x) < +∞}. The first concept of the following definition has been introduced (in a slightly different but equivalent form) in [START_REF] Jourani | Weak regularity of functions and sets in Asplund spaces[END_REF] in the case ∂ is the limiting subdifferential under the name of weakly regular function (WR) at x relative to x * . The third one has been used in [START_REF] Amahroq | Subdifferentiation and minimization of the difference of two functions[END_REF] and [START_REF] Penot | Gap continuity of multimaps, Set-Val. Anal[END_REF]. The fourth one is a variant of a notion introduced and studied in [START_REF] Ngai | ϕ-regular functions in Asplund spaces[END_REF], where ∂ = ∂ F and a quantitative estimate is required. The last one appears in [START_REF] Duda | Semiconvex and strongly paraconvex functions: representations as suprema of smooth functions[END_REF]. The other ones seem to be new.

Definition 1 (a) The function f is said to be equi-(∂)-subdifferentiable at (x, x * ), where x * ∈ ∂f (x), if there exists some ρ > 0 such that for any ε > 0 one can find some δ > 0 such that

x * , w -x ≤ f (w) -f (x) + ε w -x (1)
for all x * ∈ ∂f (x) ∩ B( x * , ρ) and all w ∈ B(x, δ).

(b) The function f is said to be boundedly equi-(∂)-subdifferentiable at x if ∂f (x) is nonempty and if for any bounded subset B * of X * and any ε > 0 one can find some δ > 0 such that (1) holds for all x * ∈ ∂f (x) ∩ B * and all w ∈ B(x, δ).

(c) The function f is said to be equi-(∂)-subdifferentiable at x if ∂f (x) is nonempty and if for any ε > 0 one can find some δ > 0 such that (1) holds for all x * ∈ ∂f (x) and all w ∈ B(x, δ).

(d) The function f is said to be uniformly ∂-subdifferentiable around (x, x * ), where x * ∈ ∂f (x), if there exists some ρ > 0 such that for any ε > 0 one can find some δ > 0 such that

x * , y -x ≤ f (y) -f (x) + ε y -x (2)
for all (x, x * , f (x)) ∈ J ∂ f ∩ B((x, x * , f (x)), ρ) and all y ∈ B(x, δ).

(e) The function f is said to be boundedly uniformly ∂-subdifferentiable around x if ∂f (x) is nonempty and if there exists some ρ > 0 such that for any bounded subset B * of X * and for any ε > 0 one can find some δ > 0 such that [START_REF] Attouch | Approximation and regularization of arbitrary functions in Hilbert spaces by the Lasry-Lions method[END_REF]

holds for all (x, x * , f (x)) ∈ J ∂ f ∩ (B(x, ρ) × B * × B(f (x), ρ)
) and all y ∈ B(x, δ).

(f ) The function f is said to be uniformly ∂-subdifferentiable around (x, f (x)) if ∂f (x) is nonempty and if there exists some ρ > 0 such that for any ε > 0 one can find some δ > 0 such that (2) holds for all

(x, x * , f (x)) ∈ J ∂ f ∩ (B(x, ρ) × X * × B(f (x), ρ)
) and all y ∈ B(x, δ).

(g) Given a subset S of X, f is said to be uniformly ∂-subdifferentiable on S if ∂f (x) is nonempty for all x ∈ S and if for any ε > 0 one can find some δ > 0 such that ( 2) holds for all

(x, x * , f (x)) ∈ J ∂ f ∩ (S × X * × R) and all y ∈ B(x, δ) ∩ S.
(h) f is said to be uniformly ∂-subdifferentiable around x if there exists some neighborhood of x on which f is uniformly ∂-subdifferentiable.

Remarks. (a) When f is equi-∂-subdifferentiable at (x, x * ) and ρ is as in assertion (a) one has ∂f (x) ∩ B( x * , ρ) ⊂ ∂ F f (x). When f is boundedly equi-∂-subdifferentiable at x, one has ∂f (x) = ∂ F f (x).
Similar assertions hold for the other notions. For such a reason, when the mention of ∂ is omitted in the sequel, that means that ∂ = ∂ F .

(b) The preceding assertions can be formulated by using hyper-modulus or modulus. Here, as usual, a modulus is a function µ : R + → R + ∪ {+∞} which is continuous at 0, with µ(0) = 0; π : R + → R ∞ is an hypermodulus if, and only if µ given by µ(t) := π(t)/t for t > 0, µ(0) = 0 is a modulus. Thus, a quantitative form can be given to the preceding notions, as in [START_REF] Ngai | ϕ-regular functions in Asplund spaces[END_REF].

Let us compare the preceding notions. The assertions of the following proposition are displayed for the convenience of the reader. They are either obvious or easy consequences of a compactness argument. Let us recall that f is said to be quiet at x ∈ dom f if -f is calm at x, i.e. if there exist c > 0 and r > 0 such that

f (x) -f (x) ≤ c x -x for all x ∈ B(x, r). We observe that ∂ F f (x) is bounded when f is quiet at x. Proposition 2 The preceding properties are ranked in increasing strength. If X is finite dimensional, f is boundedly equi-(∂)-subdifferentiable at x if, and only if, it is equi-(∂)- subdifferentiable at (x, x * ) for all x * ∈ ∂ F f (x). If f is continuous at x, f is uniformly ∂-subdifferentiable around (x, f (x)) if, and only if, f is uniformly ∂-subdifferentiable around x. If f is quiet at x, f is boundedly equi-(∂)-subdifferentiable at x if, and only if, it is equi-(∂)-subdifferentiable at x. If f is Lipschitzian around x, f is boundedly uniformly ∂-subdifferentiable around x if, and only if, it is uniformly ∂-subdifferentiable around (x, f (x)).
Example. The distinctions of Definition 1 may appear superfluous. However, even for differentiable functions, they exist. If f is differentiable at x, it is obviously equi-subdifferentiable at x but it is not necessarily uniformly subdifferentiable around x. This last property is satisfied if the derivative Df of f is uniformly continuous on some neighborhood of x.

Example. Let f : R → R be given by f (x) := √ x + , where x + := max(x, 0). Then, as noted in [22, Example 1], f is equi-∂ F -subdifferentiable at (0, 0). In fact, it is boundedly equi-∂ C -subdifferentiable at x. But it is not equi-∂ F -subdifferentiable at 0, as easily seen. It is also boundedly uniformly ∂-subdifferentiable around x = 0, but not uniformly ∂-subdifferentiable around x = 0.

Example. Suppose f admits a lower approximation s : X → R at x, i.e. that there exists a modulus µ such that f (x + u) ≥ f (x) + s(u) -u µ(u) for all u ∈ X. Then, if s is sublinear and continuous, for every x * ∈ int(∂s(0)), f is equi-∂ F -subdifferentiable at (x, x * ). Conversely, if f is equi-∂ F -subdifferentiable at (x, x * ), then f admits a lower approximation s at x which is sublinear and continuous: one can define s by s(u) := sup{ x * , u : x * ∈ ∂f (x) ∩ B( x * , ρ)}, where ρ is as in Definition 1 (a).

Obviously, convex functions satisfy all the conditions of Definition 1. In fact, property (h) of this definition corresponds to a well known class of functions. [START_REF] Duda | Semiconvex and strongly paraconvex functions: representations as suprema of smooth functions[END_REF]) Let f be continuous on some neighborhood of x. Then the following assertions are equivalent:

Theorem 3 (Duda and Zajíček

(a) f is uniformly ∂-subdifferentiable around x and f is subdifferentiable around x;

(b) there exists a neighborhood V of x such that for all ε > 0 one can find some δ > 0 such that for all x ∈ V one can find some x * ∈ X * satisfying ( 2) for all y ∈ V ∩ B(x, δ);

(c) f is semiconvex around x in the sense that there exist some neighborhood U of x and some modulus µ such that for all x, y ∈ U, t ∈ [0, 1]

f ((1 -t)x + ty) ≤ (1 -t)f (x) + tf (y) + t(1 -t)µ( x -y ) x -y .
On the other hand, property (a) of the preceding definition is satisfied under a weak convexity assumption.

Definition 4 ([27]

) A function f ∈ F (X) finite at x, is said to be approximately starshaped at x if for any ε > 0 there exists δ > 0 such that for any

x ∈ B(x, δ), t ∈ [0, 1], one has f ((1 -t)x + tx) ≤ (1 -t)f (x) + tf (x) + εt(1 -t) x -x .
(3)

Proposition 5 If f is approximately starshaped at x and if f is ∂ D -subdifferentiable at x, then f is equi-∂ D - subdifferentiable at x and ∂ D f (x) = ∂ F f (x).
Proof. Given ε > 0, let δ ∈ (0, 1) be such that relation (3) is satisfied for any x ∈ B(x, δ), t ∈ (0, 1]. Then, for all w ∈ B(x, δ), dividing by t and taking the limit inferior as (t, x) → (0 + , w), one gets

f D (x, w -x) ≤ f (w) -f (x) + ε w -x . Thus, for all x * ∈ ∂ D f (x) (1) is satisfied. The equality ∂ D f (x) = ∂ F f (x) is obvious.
Remark. When f is approximately convex at x in the sense of [START_REF] Ngai | Approximate convex functions[END_REF], one has a property which is intermediate between properties (a) and (h) of the preceding definition: for any ε > 0 one can find some δ > 0 such that (2) holds for all (x, x * , f (x))

∈ J ∂ f ∩ (B(x, δ) × X * × R) and all y ∈ B(x, δ). Proposition 6 If, for all x * ∈ ∂ F f (x), f is equi-subdifferentiable at (x, x * ), then ∂ F f (x) is locally closed. If f is equi-subdifferentiable at x, then ∂ F f (x) is weakly * closed.
Proof. Given x * ∈ ∂f (x) and ρ > 0 as in Definition 1 (a), the assertion follows from a passage to the limit in relation [START_REF] Amahroq | Subdifferentiation and minimization of the difference of two functions[END_REF]

over a sequence (x * n ) in ∂ F f (x) ∩ B( x * , ρ) converging to some x * .
When f is equi-subdifferentiable at x, one can take an arbitrary weakly * converging net in

∂ F f (x). Remark. In [22, Thm 4.2] it is stated that f is soft at x (i.e. ∂ L f (x) = ∂ F f (x)
) whenever f is equisubdifferentiable at x when X is finite dimensional. The following counter-example shows that such an assertion is not true. Example. Given a sequence (r n ) → 0 + let f : R → R be given by f (r n ) := 0 for all n ∈ N and f (x

) := x 2 for x ∈ R\{r n : n ∈ N}. Then ∂ F f (0) = {0}, so that f is equi-subdifferentiable at x but ∂ L f (0) = R, so that f is not soft at 0.
On the other hand the implication (ii)⇒(i) of [START_REF] Jourani | Weak regularity of functions and sets in Asplund spaces[END_REF]Thm 4.2] can be given a weaker form.

Proposition 7 If f is continuous at x and boundedly uniformly subdifferentiable around x, then f is soft on some neighborhood V of x. If moreover X is an Asplund space, then f is firmly regular on V in the sense that

∂ C f (x) = ∂ F f (x)
for all x ∈ V, hence is regular in the sense of [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF] that

∂ C f (x) = ∂ D f (x) for all x ∈ V.
Proof. Here ∂ := ∂ F . Let ρ > 0 be such that for any bounded subset B * of X * and for any ε > 0 one can find some δ > 0 such that (2) holds for all (x, x * , f (

x)) ∈ J ∂ f ∩ (B(x, ρ) × B * × B(f (x), ρ)) and all y ∈ B(x, δ). Let x ∈ V := B(x, ρ ), x * ∈ ∂ L f (x)
, where ρ ∈ (0, ρ] is small enough to ensure that f (w) ∈ B(f (x), ρ) for all w ∈ B(x, ρ ). Let ((x n , x * n , f (x n )) be a sequence of J ∂ f converging to (x, x * , f (x)). Taking for B * a bounded subset containing {x * n : n ∈ N}, for every ε > 0 one can find some δ > 0 such that

∀y ∈ B(x n , δ) x * n , y -x n ≤ f (y) -f (x n ) + ε y -x n
for n large enough to ensure x n -x < ρ . Passing to the limit on n in this inequality one gets

x * , y -x ≤ f (y) -f (x) + ε y -x for all y ∈ B(x, δ). Thus x * ∈ ∂ F f (x) and f is soft at x. When X is an Asplund space, ∂ C f (x) is the weak * closed convex hull of ∂ L f (x)
. Thus, every element of ∂ C f (x) satisfies the preceding relation, hence belongs to ∂ F f (x).

Stability properties

We first consider the stability of the classes defined above under usual operations.

Theorem 8 Let f = sup i∈I f i , where (f i ) i∈I is a finite family of functions which are equi-∂ D -subdifferentiable at x. Then f is equi-∂ D -subdifferentiable at x.
Proof. We first prove that for I(x) := {i ∈ I :

f i (x) = f (x)} we have ∂ D f (x) = cl * co( i∈I(x) ∂ D f i (x)).
It suffices to show that the support functions of both sides are equal, what follows from the equality

f D (x, u) = max i∈I(x) f D i (x, u),
which can be easily shown. Now, for every i ∈ I(x), ε > 0, one can find some δ i > 0 such that

∀x * i ∈ ∂ D f i (x), ∀x ∈ B(x, δ i ) x * i , x -x ≤ f i (x) -f i (x) + ε x -x .
Setting δ := min i∈I(x) δ i , taking convex combinations, one gets that

∀x * ∈ co( i∈I(x) ∂ D f i (x)), ∀x ∈ B(x, δ) x * , x -x ≤ f (x) -f (x) + ε x -x .
Taking a weak * converging net, one sees that the preceding inequality remains valid for all x * ∈ ∂ D f (x) and all x ∈ B(x, δ). Thus, f is equi-∂ D -subdifferentiable at x. The following stability result completes [START_REF] Ngai | Approximate convex functions[END_REF]Thm 3.8]. It has some similarities with [31, Cor 2] but the assumptions and the conclusions are different. Note that here h is not supposed to be convex but the qualification condition is stronger than the Robinson qualification condition.

Theorem 9 Let X and Y be Banach spaces, let W be an open subset of X and let f

:= h • g, where g : W → Y is differentiable at x ∈ W and h : Y → R ∞ is finite at y := g(x). Suppose A := Dg(x)
∂ F f (x) = ∂ F h(y) • Dg(x).
Moreover, there exists some c > 0 such that for every x * ∈ ∂ F f (x) there exists some y * ∈ ∂ F h(y) such that x * = y * • Dg(x) and y * ≤ c x * . In fact, if c > 0 is such that B Y ⊂ A(cB X ) for A := Dg(x), then for every x * ∈ A T (Y * ) there exists a unique y * ∈ Y * such that x * = y * • Dg(x) and y * ≤ c x * . The uniqueness stems from the surjectivity of A; the estimate is a consequence of the fact that for every y ∈ B Y there exists some x ∈ cB X satisfying y = A(x).

Let ρ > 0 be such that for all ε > 0 one can find some δ := δ(ε) > 0 such that y * , y -y ≤ h(y) -h(y) + ε y -y for all y * ∈ ∂ F h(y) ∩ B( y * , ρ) and all y ∈ B(y, δ). Let c > A := Dg(x) and let ρ := ρ/c . Since g is differentiable at x, given ε ∈ (0, c -A ), there exists γ

:= γ(ε) ∈ (0, δ(ε)/c ) such that g(w) -g(x) -A(w -x) ≤ ε w -x for all w ∈ B(x, γ).
Then, for all w ∈ B(x, γ) and for all

x * ∈ ∂ F f (x) ∩ B( x * , ρ ) one has g(w) -g(x) ≤ c w -x ≤ c γ ≤ δ(ε) and x * := y * • A with y * ∈ ∂ F h(y) ∩ B( y * , ρ), so that x * , w -x = y * , A(w -x) ≤ y * , g(w) -g(x) + y * .ε w -x ≤ h(g(w)) -h(g(x)) + ε g(w) -g(x) + y * .ε w -x ≤ f (w) -f (x) + ε (c + c x * + ρ) w -x .
Since ε is arbitrarily small, these inequalities show that ( 1) is satisfied. The proof of assertion (b) is similar, taking into account the fact shown above that for every bounded subset

B * of A T (Y * ) there exists a bounded subset C * of Y * such that B * = A T (C * ).
The proof of the following statement is obvious. When ∂ = ∂ L , its second hypothesis can be ensured by assuming some compactness condition and some asymptotic subdifferential condition.

Theorem 10 Let f := f 1 + f 2 , where f 1 and f 2 are equi-∂-subdifferentiable at x. Suppose ∂f (x) = ∂f 1 (x) + ∂f 2 (x). Then f is equi-∂-subdifferentiable at x.

Subdifferential characterizations

A number of classes of generalized convex functions on an Asplund space X have been given subdifferential characterizations ( [START_REF] Aussel | Subsmooth sets: functional characterizations and related concepts[END_REF], [START_REF] Colombo | Variational inequalities and regularity properties of closed sets in Hilbert spaces[END_REF], [START_REF] Daniilidis | Approximate convexity and submonotonicity[END_REF], [START_REF] Ngai | Approximately convex functions and approximately monotone operators[END_REF], [START_REF] Ngai | Paraconvex functions and paraconvex sets[END_REF]...). In the present section we examine whether the same can be done for the classes we defined above.

In this section, X is an Asplund space and ∂ stands for the Fréchet subdifferential. We say that a multimap (or multifunction) M : X ⇒ X * is semimonotone if there exists a modulus µ such that

∀(x, x * ), (y, y * ) ∈ gph(M ) x * -y * , x -y ≥ -µ( x -y ) x -y .
It is said to be submonotone at x ∈ X if there exists a modulus µ such that

∀x * ∈ M (x), (y, y * ) ∈ gph(M ) x * -y * , x -y ≥ -µ( x -y ) x -y .
We start with a qualitative reformulation of [START_REF] Ngai | ϕ-regular functions in Asplund spaces[END_REF]Thm 4] which is a result of quantitative nature.

Theorem 11 For a l.c.s. function f on X and x * ∈ ∂ F f (x) the following assertions are equivalent:

(a) f is uniformly ∂-subdifferentiable around (x, x * ); (b) there exists ρ > 0 such that, for V := B((x, x * , f (x)), ρ), the multimap M : X ⇒ X * given by

gph(M ) := {(x, x * ) : (x, x * , f (x)) ∈ J ∂ f ∩ V }, is semimonotone.
The proof of the next result is much simpler. It relies on the following version of the Mean Value Theorem.

Lemma 12 Let f be a l.c.s. function on X and let x, y ∈ dom f. Then there exist t ∈ (0, 1], sequences

(z n ) → z := x + t(y -x), (z * n ) such that z * n ∈ ∂f (z n ) for all n ∈ N and lim inf n z * n , x -y ≥ f (x) -f (y), (4)
lim inf n z * n , z -z n ≥ 0. (5)
Theorem 13 For a l.c.s. function f on X and x ∈ dom ∂f, the following assertions are equivalent: (a) f is uniformly ∂-subdifferentiable around x; (b) there exists ρ > 0 such that for V := B(x, ρ) × X * × R, the multimap M : X ⇒ X * given by

gph(M ) := {(x, x * ) : (x, x * , f (x)) ∈ J ∂ f ∩ V }, ( 6
)
is semimonotone.

Proof. (a)⇒(b) Let ρ > 0 be such that for any ε > 0 one can find some δ > 0 such that for all (x,

x * , f (x)) ∈ J ∂ f ∩ V , with V := B(x, ρ) × X * × R, and all y ∈ B(x, δ) one has x * , y -x ≤ f (y) -f (x) + ε y -x . ( 7
)
In particular, if (x, x * , f (x)) ∈ J ∂ f ∩ V and (y, y * , f (y)) ∈ J ∂ f ∩ V are such that x -y < δ, one has a similar relation by interchanging (x, x * ) with (y, y * ). Adding sides by sides the two relations, one gets

x * -y * , y -x ≤ 2ε y -x , so that M is semimonotone. (b)⇒(a)
Let ρ > 0, V and M be as in assertion (b), so that, for every ε > 0 there exists δ := δ(ε) > 0 such that for all (x, x * ), (y, y * ) ∈ gph(M ) satisfying x -y < δ one has

x * -y * , y -x ≤ ε x -y .
Given ε > 0, let δ := min(δ(ε), ρ/2). Let V := B(x, ρ/2) × X * × R. Then, for every (x, x * , f (x)) ∈ J ∂ f ∩ V and y ∈ B(x, δ ) one has y ∈ B(x, ρ). When f (y) = +∞, relation [START_REF] Borwein | The proximal normal formula in Banach space[END_REF] is trivially satisfied. Thus, we may suppose f (y) < +∞ and apply the preceding lemma: there exist t ∈ (0, 1] and sequences (z n ) → z := x + t(y -x), (z * n ) such that z * n ∈ ∂f (z n ) for all n ∈ N satisfying (4), [START_REF] Bernard | Prox-regularity of functions and sets in Banach spaces[END_REF]. Then

x * , y -x + f (x) -f (y) ≤ x * , y -x + lim inf n z * n , x -y ≤ x * , y -x + lim inf n ( x * , x -y + z * n -x * , t -1 (x -z) ) ≤ lim inf n t -1 ( z * n -x * , x -z n + z * n -x * , z n -z ) ≤ lim inf n t -1 (ε x -z n + z * n , z n -z ) ≤ t -1 ε x -z = ε x -y .
Thus, ( 7) is satisfied. Now we give a variant of [START_REF] Jourani | Weak regularity of functions and sets in Asplund spaces[END_REF]Thm 7.1]; here we use the Fréchet subdifferential instead of the limiting subdifferential and f is not supposed to be locally Lipschitzian.

Proposition 14 Let f be a l.c.s. function on X, finite at x, with ∂ F f (x) nonempty. If ∂ F f is submonotone at x, then f is equi-subdifferentiable at x. Proof. By assumption, for every ε > 0 there exists δ > 0 such that for all x * ∈ ∂ F f (x), (y, y * ) ∈ gph(∂ F f ) satisfying x -y < δ one has

x * -y * , y -x ≤ ε x -y .

Let us show that, for all x * ∈ ∂f (x) and all y ∈ B(x, δ),

x * , y -x ≤ f (y) -f (x) + ε y -x . ( 8
)
Given x * ∈ ∂f (x) and y ∈ B(x, δ) ∩ dom f, Lemma 12 yields t ∈ (0, 1], sequences (z n ) → z := x + t(y -x), (z * n ) with z * n ∈ ∂f (z n ) for all n ∈ N, such that relations (4), ( 5) hold. Then z n ∈ B(x, δ) for n large enough and, by these relations,

x * , y -x = lim n (1/t) x * , z n -x ≤ lim inf n (1/t)( z * n , z n -x + ε z n -x ) ≤ lim inf n (1/t)( z * n , z -x + ε z -x ) ≤ lim sup n ( z * n , y -x + ε y -x ) ≤ f (y) -f (x) + ε y -x .
When y ∈ B(x, δ)\ dom f, inequality ( 8) is trivial.

Nice sets and functions

We devote the present section to the passages from functions to sets. It appears that when the subjets of the functions are involved, these passages are nicer than the corresponding ones in the case of approximate functions or paraconvex functions (see [START_REF] Ngai | Approximately convex functions and approximately monotone operators[END_REF], [START_REF] Ngai | Paraconvex functions and paraconvex sets[END_REF]). We recall that the firm or Fréchet normal cone to a subset E of X at some point Proof. (a)⇒(b) Assumption (a) can be expressed as: there exists some ρ ∈ (0, 1) such that for all ε > 0 there exists δ > 0 for which

x of E is N F (E, x) := ∂ F ι E (x),
∀x ∈ E ∩ B(x, δ), ∀x * ∈ N F (E, x) ∩ B( x * , ρ) x * , x -x ≤ ε x -x . ( 9
)
Given ε > 0, let w ∈ B(x, δ/2) and let

x * ∈ ∂ F d E (x) ∩ B( x * , ρ). One can find a sequence (x n ) in E such that ( x n -w ) → d E (w) and x n -w ≤ w -x . Then x n ∈ B(x, δ) since x n -x ≤ x n -w + w -x ≤ 2 w -x . Since x * ∈ ∂ F d E (x) = N F (E, x) ∩ B X * , relation (9) ensures that x * , w -x ≤ lim sup n ( x * , x n -x + x * . x n -w ) ≤ lim sup n (ε x n -x + x n -w ) ≤ 2ε w -x + d E (w).
Since ε > 0 is arbitrary, this shows that d E is equi-subdifferentiable at (x, x * ). (b)⇒(a). Let ρ ∈ (0, 1) be such that for all ε > 0 there exists δ > 0 for which

∀w ∈ B(x, δ), ∀x * ∈ ∂ F d E (x) ∩ B( x * , ρ) x * , w -x ≤ d E (w) + ε w -x . ( 10
)
Then, for all x ∈ E ∩ B(x, δ) and for all x * ∈ N F (E, x) ∩ B( x * , ρ 2 ), we have Now let us turn to the links between a function f and its epigraph E f . For x ∈ dom f we set x f := (x, f (x)). We endow X × R with a product norm, i.e. a norm for which the projections and the insertions x → (x, 0) and r → (0, r) are nonexpansive. In both cases relation [START_REF] Bounkel | On various notions of regularity of sets in nonsmooth analysis[END_REF] is satisfied with ε changed into ε(c + 1). The proof of the next result is similar.

y * := (1 + ρ) -1 x * ∈ N F (E, x) ∩ (1 + ρ) -1 (1 + ρ 2 )B X * ⊂ ∂ F d E (x), y * -x * ≤ (1 + ρ) -1 x * -x * + ρ(1 + ρ) -1 x * ≤ (1 + ρ) -1 ρ 2 + ρ(1 + ρ) -1 = ρ, so that, by (10), y * , x -x ≤ ε x -x and x * , x -x ≤ (1 + ρ)ε x -x ≤ 2ε x -x

Theorem 19

 19 For f lower semicontinuous and x ∈ dom f, x * ∈ ∂f (x), the following assertions are equivalent: (a) The function f is equi-subdifferentiable at (x, x * );(b) The epigraph E f of f is equi-normal at (x f , ( x * , -1)).Proof. (a)⇒(b) Let ρ ∈ (0, 1) be such that for all ε > 0 one can find some δ > 0 such thatx * , w -x ≤ f (w) -f (x) + ε w -x (11) for all x * ∈ ∂f (x) ∩ B( x * , ρ) and all w ∈ B(x, δ). Let ρ E ∈ (0, 1/2) be such that 2ρ E ≤ ρ(1 + x * ) -1 and let (x * , -r) ∈ N F (E, x f ) ∩ B(( x * , -1), ρ E ). Then one has r > 1/2 and x * /r ∈ ∂ F f (x) with x * /r -x * ≤ (1/r) x * -x * + (1/r) |r -1| x * ≤ 2ρ E (1 + x * ) ≤ ρ.Thus, for all (w, r)∈ E f ∩ B(x f , δ), relation (11) yields (x * , -r), (w, r) -(x, f (x)) ≤ r ( x * /r, w -x -(f (w) -f (x))) ≤ 2ε w -x ≤ 2ε (w, r) -(x, f (x)) ,so that E f of f is equi-normal at (x f , ( x * , -1)).(b)⇒(a). Let ρ ∈ (0, 1) be such that for all ε > 0 there exists γ > 0 for whichx * , w -x -r(r -f (x)) ≤ ε (w, r) -(x, f (x)) (12) whenever (w, r) ∈ B((x, f (x)), γ) ∩ E f and (x * , -r) ∈ N F (E f , x f ) ∩ B(( x * , -1), ρ). Let x * ∈ ∂ F f (x) ∩ B( x * , ρ), so that (x * , -1) ∈ N F (E f , x f ) ∩ B(( x * , -1), ρ). Let c := x * + 1 and let δ ∈ (0, γ/(c + 1)). Let w ∈ B(x, δ). When r := f (x) + c w -x ≥ f (w) we have (w, r) ∈ B((x, f (x)), γ) ∩ E f , hence x * , w -x -(f (w) -f (x)) ≤ ε (w, r) -(x, f (x)) ≤ ε w -x + ε |r -f (x)| = ε(c + 1) w -x .When f (x) + c w -x < f (w) we havex * , w -x -(f (w) -f (x)) < x * , w -x -c w -x ≤ 0.

  where the indicator function ι E of E is defined by ι E (x) = 0 for x ∈ E, ι E (x) := +∞ for x ∈ X\E. The following results complete[START_REF] Ngai | ϕ-regular functions in Asplund spaces[END_REF] Thm 2].

Theorem 15 Let E be a closed subset of X and let x ∈ E, x * ∈ N F (E, x) ∩ B X * . The following assertions are equivalent:

(a) The set E is equi-normal at (x, x * ) in the sense that its indicator function ι E is equi-subdifferentiable at (x, x * );

(b) The distance function d E is equi-subdifferentiable at (x, x * ).

  so that[START_REF] Borwein | Techniques of Variational Analysis[END_REF] holds with ρ replaced with ρ 2 and ε replaced by 2ε.The following consequence stems from the fact that for x * ∈ N F (E, x)\{0}, E is equi-normal at (x, x * ) if, and only if, E is equi-normal at (x, u Corollary 16 For x in a closed subset E of X the following assertions are equivalent:(a) The indicator functionι E of E is equi-subdifferentiable at (x, x * ) for every x * ∈ N F (E, x); (b) The distance function d E is equi-subdifferentiable at (x, x * ) for every x * ∈ ∂ F d E (x).The proofs of the next equivalences are simpler than the preceding one since it suffices to use the relation∂ F d E (x) = N F (E, x) ∩ B X *and the fact that d E is continuous. Theorem 17 Let E be a closed subset of X and let x ∈ E. The following assertions are equivalent: (a) The set E is boundedly normal at x in the sense that its indicator function ι E is boundedly equisubdifferentiable at x; (b) The distance function d E is equi-subdifferentiable at x. Theorem 18 Let E be a closed subset of X and let x ∈ E. The following assertions are equivalent: (a) The set E is uniformly boundedly normal around x in the sense that its indicator function ι E is uniformly boundedly subdifferentiable around x; (b) The distance function d E is uniformly subdifferentiable around x.

* ) where u * := x * -1 x * .

Theorem 20 Let f be lower semicontinuous and let x ∈ dom f. The following assertion (b) implies assertion (a). If f is quiet at x, both assertions are equivalent:

(a) The function f is boundedly equi-subdifferentiable at x; (b) The epigraph E f of f is boundedly equi-normal at x f .

Proof. We just mention the necessary changes. (b)⇒(a) Given a bounded subset B * of X * and ε > 0 one can find γ > 0 such that

and let δ := γ/(c + 1). Then, as in the preceding proof, for every x * ∈ ∂ F f (x) ∩ B * and for every w ∈ B(x, δ), we get that relation [START_REF] Bounkel | On various notions of regularity of sets in nonsmooth analysis[END_REF] is satisfied with

Let A * be a subset of N F (E f , x f ) contained in the ball with center (0, 0) and radius a > 0. Since A * is contained in Q 0 , for every (u * , -r * ) ∈ A * \{(0, 0)} we have r * > 0 and x * := u * /r * ∈ ∂ F f (x) ⊂ cB X * . Thus, for all ε > 0, one can find some δ > 0 such that

for all (u * , -r * ) ∈ A * , all w ∈ B(x, δ) and all r ≥ f (w).

The proof of the next result is not as simple; we refer to [START_REF] Ngai | ϕ-regular functions in Asplund spaces[END_REF]Thm 3].

Now let us consider the case of sublevel sets.

Proposition 22 Let X be an Asplund space and let f :

Proof. We can pick c > 0, r > 0 such that x * ≥ c for all x ∈ (X\S) ∩ B(x, r) and all x * ∈ ∂ F f (x). Then, by [START_REF] Penot | Well-behavior, well-posedness and nonsmooth analysis[END_REF]Thm 9.1] with ϕ = c, (see also, among several other contributions, [START_REF] Cornejo | Conditioning and upper-Lipschitz inverse subdifferentials in nonsmooth optimization problems[END_REF], [START_REF] Ngai | Metric inequalities, subdifferential calculus and applications[END_REF], [START_REF] Zhang | Upper-Lipschitz multifunctions and inverse subdifferentials[END_REF], with various assumptions on X) we have f + (x) ≥ cd S (x) for x ∈ V, where f + := max(f, 0) and

Elementary calculus rules yield

Given w ∈ S ∩ B(x, δ), x * ∈ ∂ D d S (x), by the preceding inclusion and inequality and a passage to the convex hull and the closure, we get cx * , w -x ≤ cε w -x . Thus, d S is equi-∂ D -subdifferentiable at x, hence S is equi-normal at x by Theorem 20.