Differentiability of quasiconvex functions

Differentiability of quasiconvex functions on separable Banach spaces

Patrick J. Rabier Department of mathematics, University of Pittsburgh, Pittsburgh, PA 15260 rabier@imap.pitt.edu
Abstract.

We investigate the differentiability properties of a real-valued quasiconvex function defined on a separable Banach space Continuity is only assumed to hold at the points of a dense subset. If so, this subset is automatically residual.

Sample results that can be quoted without involving any new concept or nomenclature are as follows: (i) If is usc or strictly quasiconvex, then is Hadamard differentiable at the points of a dense subset of (ii) If is even, then is continuous and Gâteaux differentiable at the points of a dense subset of In (i) or (ii), the dense subset need not be residual but, if is also reflexive, it contains the complement of a Haar null set. Furthermore, (ii) remains true without the evenness requirement if the definition of Gâteaux differentiability is generalized in an unusual, but ultimately natural, way.

The full results are much more general and substantially stronger. In particular, they incorporate the well known theorem of Crouzeix, to the effect that every real-valued quasiconvex function on is Fréchet differentiable a.e.

Key words and phrases:
1991 Mathematics Subject Classification:

1. Introduction

According to Rockafellar [38, p. 428], it has been known since the early 20th century that a real-valued convex function on is everywhere continuous and a.e. Fréchet differentiable, although the geometric leanings prevalent at that time make it difficult to pinpoint the origin of this result with greater accuracy.

When is replaced by an infinite dimensional Banach space a real-valued convex function on is either continuous at every point or discontinuous at every point. This makes it obvious that the investigation of the differentiability properties of convex functions should be limited to the former class. When is separable, Mazur [31] proved the residual Gâteaux differentiability of continuous convex functions in 1933. Recall that, in Baire category terminology, a residual subset is the complement of a set of first category. Much later, in 1976, Aronszajn [1] showed that Gâteaux differentiability is also true almost everywhere, provided that a suitable generalization of null sets is used when (Aronszajn null sets; see Subsection 2.1).

In the theorems of Mazur and Aronszajn, Gâteaux differentiability can be replaced by Hadamard differentiability -and hence by Fréchet differentiability when - since both concepts coincide for locally Lipschitz functions (this is well known [42, p. 19] and elementary). More generally, the residual Fréchet differentiability of continuous convex functions was proved in a 1968 landmark paper by Asplund [2] when is separable, as well as in other cases that do not require or imply the separability of

The dichotomy everywhere/nowhere continuous is no longer true for finite quasiconvex functions, where the quasiconvexity of is understood as for every and every Nonetheless, in 1981, Crouzeix [14] (see also [9]) proved that, just like convex functions, real-valued quasiconvex functions on not necessarily continuous, are a.e. Fréchet differentiable. To date, this property has not been extended in any form to infinite dimensional spaces. In fact, with only a few notable exceptions, the literature has consistently focused on differentiability under local Lipschitz continuity conditions, a topic with a long history which is still the subject of active research; see the recent text [25]. Of course, local Lipschitz continuity is grossly inadequate to handle quasiconvex functions, even continuous ones, irrespective of

It is the purpose of this paper to show how Crouzeix’s theorem can be generalized when is replaced by a separable Banach space We provide variants of the theorems of Mazur and Aronszajn in the more general setting of quasiconvex functions, although neither theorem has a genuine generalization. In particular, “mixed” criteria must be used to evaluate the size of the set of points of differentiability. We shall return to this and related issues further below.

As a simple first step, recall that the set of points of discontinuity of any real-valued function on a topological space is an (see e.g. [21, p.78], [39, p. 58]), so that either this set is of first category, or it has nonempty interior111It is only in Baire spaces (e.g., Banach spaces) that these two options are mutually exclusive.. Evidently, no generic differentiability result should be expected in the latter case, which dictates confining attention to quasiconvex functions that are continuous at the points of a residual subset of the (separable) Banach space Incidentally, Mazur’s theorem breaks down for this class of functions, even when and continuity is assumed. Indeed, in general, a monotone continuous function on is only differentiable at the points of a set of first category ([44, Corollary 1]), even though its complement has measure by Lebesgue’s theorem. As we shall see later, Aronszajn’s theorem also fails in the quasiconvex (even continuous) case when

In Banach spaces, residual sets are dense and the converse is trivially true for sets. In particular, a real-valued function is continuous at the points of a residual subset if and only if it is  densely continuous, i.e.,  continuous at the points of a dense subset. Common examples of densely continuous functions include the upper or lower semicontinuous functions ([32, Lemma 2.1]) and, by a theorem that goes back to Baire himself, the so-called functions of Baire first class, i.e., the pointwise limits of sequences of continuous functions ([43, p. 12]). An apparently new class (ideally quasiconvex functions) that contains all the semicontinuous quasiconvex functions -and even all the quasiconvex functions when - is described in Subsection 3.2.

As shown by the author in [37], when is a Baire topological vector space (tvs), the densely continuous quasiconvex functions on have quite simple equivalent characterizations in terms of the lower level sets

(1.1)

where Of course, since is real-valued, but the notation will occasionally be convenient. By the quasiconvexity of all the sets and are convex and, conversely, if all the (or ) are convex, then is quasiconvex, which is de Finetti’s original definition [19].

In [37] and again in this paper, a special value plays a crucial role, which is the so-called  topological essential infimum of denoted by for simplicity, defined by

(1.2)

The last equality in (1.2) follows from the fact that the sets are linearly ordered by inclusion. Since is real-valued and is a Baire space, it is always true that Note also that and that the set (but not ) is always of first category since it is the union of countably many with

From now on, is a separable Banach space and is quasiconvex and densely continuous. Below, we give a synopsis of the differentiability results proved in this paper. The rough principle is that the size of the subset ranks the amount of differentiability of (the smaller the better). Even though is always of first category, various refinements will be involved. Many of the concepts and some technical results needed for the proofs are reviewed or introduced in the next two sections. In particular, several properties of with direct relevance to the differentiability question are established in Section 3.

In Section 4, we focus on the differentiability of above level that is, at the points of The special features of make it possible to rely on a theorem of Borwein and Wang [7] to prove that is Hadamard differentiable on except at the points of an Aronszajn null set (Theorem 4.2). If so that this fully generalizes Aronszajn’s theorem and settles the differentiability issue.

Accordingly, in the remainder of the discussion, is assumed. It turns out that is always semi-open, i.e., contained in the closure of its interior (which is not true when is replaced by an arbitrary convex set), so that the differentiability result just mentioned above still shows that is Hadamard differentiable at “most” points of That may be empty does not invalidate this statement. Hence, it only remains to investigate differentiability at the points of

It turns out that is never Gâteaux differentiable at any point of (Theorem 5.1). Thus, the differentiability of on depends solely upon its differentiability at the points of and, if is Gâteaux differentiable at then (Theorem 5.1), as if were a genuine minimum. In summary, the problem is to evaluate the size of those points of at which the directional derivative of exists and is in all directions and to decide whether is Hadamard differentiable at such points. While this demonstrates the importance of in the differentiability question, the task is not as simple as one might first hope.

When is not only of first category but also nowhere dense because, in finite dimension, convex subsets of first category are nowhere dense. Differentiability when is separable and is nowhere dense is the topic of Section 5. The technicalities depend upon whether is of first or second category but, ultimately, we show that is Hadamard -but not Fréchet- differentiable on the complement of the union of a nowhere dense set with an Aronszajn null set (Theorem 5.4). Such unions cannot be replaced with Aronszajn null sets alone or with sets of first category alone.

Every subset which is the union of a nowhere dense set with an Aronszajn null set has empty interior (of course, this is false if “nowhere dense” is replaced by “first category”) and the class of such subsets is an ideal, but not a -ideal. In particular, above -or any finite collection of similar functions- is Hadamard differentiable at the points of a dense subset of This is only a “subgeneric” differentiability property but, if is separable and reflexive, a truly generic result holds: If is nowhere dense, is Hadamard differentiable on except at the points of a Haar null set (Theorem 5.5). This should be related to the rather unexpected fact that, in such spaces, a quasiconvex function is densely continuous if and only if its set of points of discontinuity is Haar null (Theorem 3.3).

Examples show that Theorem 5.5 is not true for Fréchet differentiability, or if is not reflexive, or if “Haar null” is replaced by “Aronszajn null”, even if is continuous and Thus, Aronszajn’s theorem cannot be extended to (densely) continuous quasiconvex functions if Nonetheless, when Crouzeix’s theorem is recovered because every quasiconvex function on is densely continuous, the Haar null sets of are just the Borel subsets of Lebesgue measure and Hadamard and Fréchet differentiability coincide on

Aside from there are several conditions ensuring that is nowhere dense. In particular, if is upper semicontinuous (usc) or, more generally, if for then Also, is nowhere dense if is strongly ideally quasiconvex (Definition 3.1, Theorem 3.4) or strictly quasiconvex and densely continuous (Corollary 5.6).

In general, need not be nowhere dense if especially when is lower semicontinuous (lsc). Thus, loosely speaking, the differentiability issue is more delicate for lsc functions than for usc ones. However, no difficulty arises as a result of passing to the lsc hull, even though doing so can only enlarge In Subsection 5.2, we show that Theorems 5.4 and 5.5 are applicable to a densely continuous quasiconvex function if and only if they are applicable to its lsc hull.

The case when is not nowhere dense is discussed in Section 6. This is new territory since it never happens when or when is usc, let alone convex and continuous. We show that a subset of often much smaller than is intimately related to the Gâteaux differentiability question. Specifically, consists of all the limits of the convergent sequences of points of that remain in some finite dimensional subspace of

In a more arcane terminology, is the sequential closure of for the finest locally convex topology on This type of closure has been around for quite a while in the literature, but only in connection with issues far removed from differentiability (topology, moment problem in real algebraic geometry, etc.) and often in a setting that rules out infinite dimensional Banach spaces (countable dimension).

In Theorem 6.2, we prove that if has empty interior, then is both continuous and Gâteaux differentiable at the points of a dense subset of Since Hadamard differentiability implies continuity, this is weaker than the analogous result when is nowhere dense (Theorem 5.4), but applicable in greater generality.

Corollaries are given in which the assumption that has empty interior is replaced by a more readily verifiable condition. Most notably, we show in Corollary 6.5 that every even densely continuous quasiconvex function on a separable Banach space is continuous and Gâteaux differentiable at the points of a dense subset of Furthermore, if is also reflexive, is continuous and Gâteaux differentiable on except at the points of a Haar null set. These properties remain true when exhibits more general symmetries (Corollary 6.6).

Section 7 is devoted to an example showing that when is not nowhere dense, the differentiability properties of at the points of or, equivalently, are indeed significantly weaker than when is nowhere dense. This confirms that an optimal outcome cannot be obtained without splitting the investigation of the two cases.

The results of Section 6, particularly those incorporating some symmetry assumption about make it legitimate to ask whether always has empty interior. If true, this would imply that Theorem 6.2 is always applicable. In Section 8, we put an early end to this speculation, by exhibiting a convex subset of of first category such that The construction of also shows how to produce densely continuous quasiconvex functions such that and but we were unable to determine whether any such function fails to be Gâteaux differentiable at the points of a dense subset.

The aftermath of not always having empty interior is that, when it remains an open question whether every densely continuous quasiconvex function on a separable Banach space is continuous and Gâteaux differentiable at the points of a dense subset. Nevertheless, in Section 9, this question is answered in the affirmative after the definition of Gâteaux differentiability is slightly altered.

A basic remark is that the concept of Gâteaux derivative at a point continues to make sense if it is only required that the directional derivatives at exist and are represented by a continuous linear form for some residual set of directions in the unit sphere. This suffices to define an “essential” Gâteaux derivative at in a unique way, independent of the residual set of directions (which is not true if “residual” is replaced by “dense”). With this extended definition, the problem is resolved in Theorem 9.2. In particular, if is reflexive and separable, every densely continuous quasiconvex function is continuous and essentially Gâteaux differentiable on except at the points of a Haar null set.

2. Preliminaries

We collect various definitions and related results that will be used in the sequel. Most, but not all, of them are known, some more widely than others. Whenever possible, references rather than proofs are given.

2.1. Aronszajn null sets

Let denote a separable Banach space. If is any sequence, call the class of all Borel subsets such that where is a Borel “null set on every line parallel to ”, that is, for every where is the one-dimensional Lebesgue measure. The Aronszajn null sets ([1]) are the (Borel) subsets of such that for every sequence such that

The Aronszajn null sets form a -ring that does not contain any nonempty open subset and every Borel subset of an Aronszajn null set is Aronszajn null. When they are the Borel subsets of Lebesgue measure It was shown by Csörnyei [16] that the Aronszajn null sets coincide with the Gaussian null sets (Phelps [34]; these are the Borel subsets of that are null for every Gaussian222For every the measure on defined by has a Gaussian distribution. probability measure on ) and also with the “cube null” sets of Mankiewicz [26].

Aronszajn null sets are preserved by affine diffeomorphisms of i.e., translations and bounded invertible linear transformations. As noted by Matous̆ková [28], if is a convex subset of with nonempty interior, the boundary is Aronszajn null, but this need not be true if even if is closed and is Hilbert.

2.2. Haar null sets

Let once again denote a separable Banach space. The Borel subset is said to be Haar null if there is a Borel probability measure on such that for every This definition is due to Christensen [10], [11]; see also [29]. It is obvious that every Aronszajn null set is Haar null, but the converse is false [34], except when The Haar null sets also form a -ring (though this is not obvious from just the definition) containing no nonempty open subset, every Borel subset of a Haar null set is Haar null and, just like Aronszajn null sets, they are preserved by affine diffeomorphisms.

In practice, it is not essential that above be a probability measure. The definition is unchanged if is a Borel measure having the same null sets as a probability measure. For example, this happens if where and is the Lebesgue measure on (change into with ).

If, in addition, is reflexive, every closed convex subset of with empty interior is Haar null (Matous̆ková [30]) but, as noted in the previous subsection, not necessarily Aronszajn null.

2.3. Gâteaux and Hadamard differentiability

A real-valued function on an open subset of a Banach space is Gâteaux differentiable at if there is such that for every  If so, is denoted by If, in addition, the limit is uniform for in every compact subset of then is said to be Hadamard differentiable at Equivalently, is Hadamard differentiable at if and only if for every sequence tending to and every sequence tending to in If is Hadamard differentiable at then it is continuous at This is folklore (see e.g. [3]) and elementary. Of course, Gâteaux differentiability alone does not ensure continuity. When it is plain from the first definition that Hadamard differentiability and Fréchet differentiability coincide.

2.4. Cone monotone functions

Let be a Banach space and be a closed convex cone with nonempty interior. If is a nonempty open subset, a function is said to be -nondecreasing if and imply This concept has long proved adequate to extend Lebesgue’s theorem on differentiation of monotone functions. It goes back (in ) to the 1937 edition of the book by Saks [40] and resurfaced in the work of Chabrillac and Crouzeix [9]. Its use in separable Banach spaces, by Borwein et al. [5] and Borwein and Wang [7], is more recent. The following result is essentially [7, Theorem 18].

Theorem 2.1.

(Borwein-Wang). Let be a separable Banach space, be a closed convex cone with and be a nonempty open subset. Suppose that is -nondecreasing. Then is Hadamard differentiable on except at the points of an Aronszajn null set.

In [7], Theorem 2.1 is proved only when but, as shown below, it is not difficult to obtain the general case as a corollary. If we use the notation if and if These relations are obviously transitive since is stable under addition (but is an ordering if and only if this is unimportant).

First, in Theorem 2.1 is locally bounded (above and below) on To see this, let be given and choose In particular, After replacing by for small enough, which does not affect it follows from the openness of that we may assume that and If and with small enough, then and so that and so that Therefore, which proves the claim.

From the above, for every there is an open neighborhood of such that is bounded on and, by the separability of there is a covering of by countably many Thus, since a countable union of Aronszajn null sets is Aronszajn null and since implies that is -nondecreasing on it is not restrictive to prove Theorem 2.1 with replaced by or, equivalently, to prove it under the additional assumption that is bounded above and below on This can be done by using Theorem 2.1 with and replaced by a finite -nondecreasing extension of to Such an extension can be obtained as follows: Since is bounded on let be such that for every and set

Since and is transitive, it is straightforward to check that is -nondecreasing on that on and that so that is real-valued (when the set is not empty, so that the supremum in the definition of is never ). Other extensions are described in [7], but they need not be finite when is finite.

Remark 2.1.

The Borwein-Wang theorem was refined by Duda [18, Remark 5.2]: is Hadamard differentiable on except at the points of a subset in a class introduced by Preiss and Zajíc̆ek, which is strictly smaller than the class of Aronszajn null sets [36, Proposition 13]. The class may replace the Aronszajn null sets everywhere in this paper.

2.5. Convex sets and category

Convex sets have a few special properties relative to Baire category. Two useful ones are given below. Like several other results, they are stated in Banach spaces but are valid in greater generality.

Lemma 2.2.

([37, Lemma 3.1]) Let be a Banach space and let and be subsets of with open and convex. If is of first category and then

The next lemma is a by-product of Lemma 2.2.

Lemma 2.3.

([37, Lemma 3.2]) If is a Banach space and is convex, then is locally of Baire second category in that is, for every open subset such that is nonempty, is of second category in

Of course, the spirit of Lemma 2.3 is that even locally, the exterior of a convex subset is always a large set in some sense. While intuitively clear when the existence of dense and convex proper subsets of infinite dimensional Banach spaces makes this issue less transparent in general.

3. Densely continuous quasiconvex functions

In the Introduction, we defined the densely continuous real valued functions on a tvs to be those functions that are continuous at the points of a dense subset of Recall that when is a Baire space, for instance a Banach space, this is actually equivalent to continuity at the points of a residual subset of

3.1. Some general properties

The first theorem of this section is part of the main result of [37]. It gives a characterization of the densely continuous quasiconvex functions in terms of their lower level sets. The notation (1.1) is used and will be used throughout the paper.

Theorem 3.1.

([37, Corollary 5.3]) Let be a Banach space and let be quasiconvex. Set (see (1.2)). Then, is densely continuous if and only if (i) when and (ii) either or and is nowhere dense when

Other conditions equivalent to, or implying, dense continuity for quasiconvex functions are given in [37], but Theorem 3.1 will suffice for our purposes. Since “first category”, “nowhere dense” and “empty interior” are synonymous for convex subsets of Theorem 3.1 shows that every real-valued quasiconvex function on is densely continuous. This argument is independent of Crouzeix’s theorem mentioned in the Introduction.

In the next theorem, we prove a number of properties of the set which will be instrumental in the discussion of the differentiability question. Part (i) was already noticed in [37]. The very short proof is given for convenience.

Theorem 3.2.

Let be a Banach space and let be quasiconvex and densely continuous. Set
(i) If (i.e., ) is of first category, then (and hence also ) is nowhere dense.
(ii) If (i.e., ) is of second category, then In particular, is nowhere dense. If is separable, is Aronszajn null.
(iii) is semi-open, i.e., contained in the closure of its interior
(iv) If is separable, the set of points of discontinuity of in is contained in an Aronszajn null set.
(v) If is separable, is contained in an Aronszajn null (Haar null) set if and only if is Aronszajn null (Haar null).

Proof.

(i) If then is not continuous at so that where is the set of points of discontinuity of Since is densely continuous, is of first category, whence is of first category, that is, is nowhere dense.

(ii) Since is of second category, On the other hand, if and is continuous at then Since the set of points of discontinuity of is of first category, this means that In particular, so that, by Lemma 2.2, Thus, and so

A convex set with nonempty interior has the same boundary as its closure ([4, p. 105]) and the boundary of a closed set is nowhere dense. Thus, is nowhere dense. Since and is convex, is Aronszajn null when is separable (Subsection 2.1).

(iii) When this does not follow from the convexity of alone (the complement of a dense convex subset is not semi-open) but, by (i) and (ii), we also know that either is nowhere dense, or

If is nowhere dense, is dense in and therefore in Suppose now that The only points of which are not already in lie on As just pointed out in the proof of (ii), and have the same boundary because is convex and Therefore, if and then is in the closure of

(iv) As before, let denote the set of points of discontinuity of The claim is that is contained in an Aronszajn null set. If then and there are a sequence with and such that either for every or for every With no loss of generality, we may assume that and, since that (of course, this is redundant when ).

In the first case (i.e., ), but so that In the second (i.e., ), but so that once again Thus, By Theorem 3.1, when so that is Aronszajn null since is convex (Subsection 2.1) and so is Aronszajn null.

(v) The sufficiency is trivial. To prove the necessity, suppose then that is contained in an Aronszajn null (Haar null) set. As noted in the proof of (i) above, where is the set of points of discontinuity of and so By (iv), is contained in an Aronszajn null set, so that is Aronszajn null (Haar null) since it is Borel.    

While the quasiconvexity of is actually unnecessary in part (i) of Theorem 3.2, it is important in part (ii). By Theorem 3.1, every lower level set -not just - is either nowhere dense or has nonempty interior, but it is only when that both options are possible ( is nowhere dense when since with and has nonempty interior when ). In contrast, is always of first category but it may or may not be nowhere dense.

Remark 3.1.

It is readily checked that the complement of an Aronszajn null set in a semi-open subset is dense in that subset. Thus, when is separable, parts (iii) and (iv) of Theorem 3.2 show that the set of points of continuity of in is dense in

In a separable and reflexive Banach space, a finite quasiconvex function is densely continuous if and only if its set of points of discontinuity is Haar null. This perhaps surprising result is proved next.

Theorem 3.3.

Let be a reflexive and separable Banach space. The quasiconvex function is densely continuous if and only if the set of points of discontinuity of is Haar null.

Proof.

Since the complement of a Haar null set is dense, the “if” part is clear. Conversely, since the set of points of discontinuity of is an (hence Borel), it suffices to show that if is densely continuous, is contained in a Haar null set. This will be seen by a suitable modification of the proof of part (iv) of Theorem 3.2.

If there are a sequence with and such that either for every or for every With no loss of generality, we may assume that and

In the first case, but so that and, in the second, but so that once again If ( and) it follows from Theorem 3.1 that has empty interior. Since is reflexive and separable, is Haar null (Subsection 2.2). If then by Theorem 3.1, so that is Aronszajn null (Subsection 2.1) and therefore Haar null. Thus, is Haar null and Since is Borel -indeed an - it is Haar null.    

When the set of points of discontinuity of is even -porous; see [7, Theorem 19] (as pointed out in [37, Remark 5.2], the lsc assumption in that theorem is not needed).

3.2. Ideally quasiconvex functions

The subset of the Banach space is said to be ideally convex if for every bounded sequence and every sequence such that The boundedness of ensures that is absolutely convergent. Such subsets were introduced by Lifšic [24] in 1970 but, apparently, they have not been used in connection with quasiconvex functions. Indeed, without Theorem 3.1, the purpose of doing so is not apparent.

Definition 3.1.

The function is ideally quasiconvex (strongly ideally quasiconvex) if its lower level sets () are ideally convex.

It is readily seen that a strongly ideally quasiconvex function is ideally quasiconvex (use ), but the converse is false; see Remark 7.1 later. Also, is ideally quasiconvex if and only if for every bounded sequence and every sequence such that Strong ideal convexity amounts to whenever for every This has nothing to do with strict quasiconvexity. For instance, every usc quasiconvex function is strongly ideally quasiconvex (see below).

If is ideally convex, it is CS-closed as defined by Jameson [22] and the two concepts coincide when is bounded. By a straightforward generalization of Carathéodory’s theorem (see for example [13]), every convex subset of is ideally convex. Therefore, every quasiconvex function on is strongly ideally quasiconvex. Open convex subsets are CS-closed ([20]), hence ideally convex, and the same thing is trivially true of closed convex subsets. In particular, lsc (usc) quasiconvex functions are ideally quasiconvex (strongly ideally quasiconvex).

We shall also need the remark that an ideally convex subset can only be nowhere dense or have nonempty interior. Indeed, if is not nowhere dense, is dense in for some nonempty open ball Since and are ideally convex, is ideally convex (obvious) and bounded and therefore CS-closed. By [22, Corollary 1], this implies that so that and so has nonempty interior.

The main properties of (strongly) ideally quasiconvex functions are captured in the next theorem.

Theorem 3.4.

Let be a Banach space.
(i) If is ideally quasiconvex, then is quasiconvex and densely continuous.
(ii) If is strongly ideally quasiconvex and then is quasiconvex and densely continuous and is nowhere dense. More generally, this is true if is ideally quasiconvex and is ideally convex.

Proof.

(i) If then is ideally convex but not nowhere dense since is of second category by definition of Thus by the remark before the theorem. Now, let and, by contradiction, suppose that is not nowhere dense, so that the larger (ideally convex) is not nowhere dense either. By the remark before the theorem, and so for every But is of first category by definition of if and a contradiction arises. Thus, is nowhere dense when That is densely continuous now follows from Theorem 3.1.

(ii) By (i), is densely continuous. Since is ideally convex and has empty interior (it is of first category), it is nowhere dense, once again by the remark before the theorem.    

Strongly ideally quasiconvex functions are the closest generalization of finite dimensional quasiconvex functions. As we shall see in Section 5, both have the same differentiability properties. By Theorem 3.4 (i), ideally quasiconvex functions possess only the continuity properties of the finite dimensional case.

4. Differentiability above the essential infimum

Throughout this section, is a separable Banach space and is quasiconvex and densely continuous. With the goal is to prove that is Hadamard differentiable at every point of except at the points of an Aronszajn null set.

The last preliminary lemma will enable us to use Theorem 2.1 to settle the differentiability question at points of The line of argument of the proof, but with other assumptions, has been used before ([15, Theorem 3.1], [7, Proposition 2]).

Lemma 4.1.

Let be a point of continuity of Then, and there is an open neighborhood of contained in and a closed convex cone with nonempty interior such that is -nondecreasing on

Proof.

Since choose so that by Theorem 3.1. Pick and small enough that that (this is possible since and is continuous at in particular, is contained in the interior of ) and that Now, set so that is the midpoint of and and

Clearly, is a convex cone with nonempty interior. That it is closed easily follows from the assumption Also, if then where is chosen such that Hence, if it follows that

Suppose now that and From the above, By writing for some this yields Since is quasiconvex, the latter because (recall and ).    

Theorem 4.2.

Let be a separable Banach space and let be quasiconvex and densely continuous. Set Then, is Hadamard differentiable on except at the points of an Aronszajn null set.

Proof.

With no loss of generality, assume that Since is dense in (Theorem 3.2 (iii)), it follows that If denotes the set of points of continuity of in then by Lemma 4.1 and since is open and nonempty and is densely continuous.

If it follows from Lemma 4.1 and Theorem 2.1 that there is an open neighborhood of contained in such that is Hadamard differentiable on except at the points of an Aronszajn null set. Since is separable, the open set is Lindelöf and therefore coincides with the union of countably many As a result, is Hadamard differentiable on except at the points of an Aronszajn null set.

Lastly, since contains all the points of continuity of in the points of are the points of discontinuity of in By Theorem 3.2 (iv), these points are contained in an Aronszajn null set, so that, as claimed, is Hadamard differentiable on except at the points of an Aronszajn null set.    

A variant of Remark 3.1 may be repeated: Since is semi-open (Theorem 3.2 (iii)), Theorem 4.2 implies that is Hadamard differentiable at the points of a dense subset of The next corollary is obvious and settles the case when

Corollary 4.3.

Let be a separable Banach space and let be quasiconvex and densely continuous. If then is Hadamard differentiable on except at the points of an Aronszajn null set.

5. Hadamard differentiability when is nowhere dense

Since the case when