1 Introducction

In this article, we take as reference the classic book Gâteaux Differentiability of Convex Functions and Topology: Weak Asplund Spaces ^{1}, in which Marián J. Fabian talks about the Asplund spaces. First, we define the multivalued functions as follows: Let 𝑋 and 𝑌 be sets. A multivalued function from 𝑋 to 𝑌 is a relation which assigns to each 𝑥 ( 𝑋 a subset of 𝑌, denoted 𝑓(𝑥); 𝑓(𝑥) = ø for some 𝑥 ∊ 𝑋 is admitted. The graph of the multivalued functions 𝑓 is the set Graph(𝑓): = {(𝑥, 𝑦) ∊ 𝑋 ( 𝑌: 𝑦 ( 𝑓(𝑥)}. If for each 𝑥 ∊ 𝑋, 𝑓(𝑥) has only one element, we say the function is single-valued. A single-valued function from 𝑋 to 𝑌, is a relation 𝑥 ↦ 𝑦 = 𝑓(𝑥) which assigns to each 𝑥 ∊ 𝑋 a unique 𝑦 ∊ 𝑌. The graph of such functions is Graph(𝑓): = {(𝑥, 𝑦) ∊ 𝑋 ( 𝑌: 𝑦 = 𝑓(𝑥)}.

The objective of this article is the study of the properties of the usco functions. A usco function is a multivalued function 𝑓: 𝑋 → 𝑌 between Banach spaces which is upper semicontinuous and 𝑓(𝑥) is compact for each 𝑥 ∊ 𝑋. A usco function 𝑓: 𝑋 → 𝑌 is minimal if for each usco function 𝑔: 𝑋 → 𝑌 such that

One of the most important examples of the usco functions appears in the study of differentiability of the convex functions defined on an open convex subset Ω of a Banach space 𝑋. It is convenient to note that the subdifferential of a function 𝑓: Ω → ℝ ∪ {+∞} in 𝑥 ∊ Ω is the multivalued function ∂𝑓: 𝑋 → 𝑋´ defined as follows:

It is also important to observe that in the completely regular topological spaces, singletons and closed sets can be separated by open sets. If 𝑋 is a Banach space, the topological dual with the weak topology is a completely regular space. Let us remember that a subset 𝐺 of a topological space 𝑋 is residual if there exists a countable family

The Asplud 𝐺-spaces are Banach spaces in which every continuous convex function, defined on an open convex set Ω, is differentiable in the Gâteaux sense in a residual subset of Ω. We denote by

2 𝛽-Differentiability

Let us start with the following definition:

**Definition 2.1** Let 𝑋 be a vector space on a field 𝕂. A vector bornology in 𝑋 is a subset 𝛽 of parts of 𝑋, denoted by ℘(𝑋), satisfying the following axioms:

𝑏𝑜𝑟𝑣 1. The union of all sets of 𝛽 is 𝑋:

𝑏𝑜𝑟𝑣 2. 𝛽 is stable under inclusions. That is, if 𝐵 ( 𝛽 and 𝐷 ⊂ 𝐵, then 𝐷 ( 𝛽.

𝑏𝑜𝑟𝑣 3. 𝛽 is stable under finite union. That is, if

𝑏𝑜𝑟𝑣 4. 𝛽 is stable under the sum operation. That is, if

𝑏𝑜𝑟𝑣 5. 𝛽 is stable under the scalar multiplication operation. That is, if 𝑆 ∊ 𝛽 and λ ( 𝕂, then λ𝑆 ∊ 𝛽.

𝑏𝑜𝑟𝑣 6 is stable under the formation of balanced envelope. That is, if 𝑆 ∊ 𝛽, then 𝑏𝑎𝑙 (𝑆) ∊ 𝛽. Where 𝑏𝑎𝑙 (𝑆) denote the balanced hull of a set 𝑆 defined by

In the context of a topological vector space (𝑋, τ), we have some natural bornologies.

**Examples 2.2** Let (𝑋, τ) be an 𝑡.𝑣.𝑠., then the following collections of subsets of 𝑋 are vector bornologies.

1. The 𝐹-bornology

2. The 𝐻-bornology

3. The 𝐺-bornology

From the definitions of these bornologies, it is easy to see that

For a proof of this result see ^{2}.

In the sequel, 𝑋 and 𝑌 denote Banach spaces, Ω is an open subset of 𝑋 and ℒ(𝑋, 𝑌) denotes the space of linear transformation from 𝑋 to 𝑌 that are continuous.

**Definition 2.3** Let 𝛽 be a bornology in 𝑋. We say that a function 𝑓: Ω → 𝑌 is 𝛽-differentiable in 𝑎 ∊ Ω if there exists a function 𝑢 ∊ ℒ(𝑋, 𝑌) such that for every 𝑆 ∊ 𝛽

The linear and continuous function 𝑢 is called 𝛽-derivative of 𝑓 at the point 𝑎 ∊ Ω and is denoted by

The following result provides a differentiability criterium with respect to a given bornology:

**Theorem 2.4** Let 𝑋 and 𝑌 be 𝑡.𝑣.𝑠., Ω an open subset of 𝑋, and 𝛽 be a bornology in 𝑋. A necessary condition for 𝑓: Ω → 𝑌 to be 𝛽-differentiable in 𝑎 ∊ Ω is that for every 𝑆 ∊ 𝛽

This result follows doing a suitable modification in the proof of proposition 1.23 in ^{3}.

In the case of a convex continuous functions 𝑓:Ω ⊂ 𝑋 → 𝑌, we have the following characterization of the 𝛽-differentiable in 𝑎 ∊ Ω:

**Theorem 2.5** Let 𝑋 be an 𝑡.𝑣.𝑠., Ω an open convex subset of 𝑋, and 𝛽 be a bornology in 𝑋. A necessary and sufficient condition for a convex and continuous function 𝑓:Ω → ℝ be 𝛽-differentiable in 𝑎 ∊ Ω is that for every 𝑆 ∊ 𝛽

Similar considerations as theorem 2.4.

2.1 Single-Valued Maps Derivatives

Now, we will study the main notions of differentiation in a 𝑡.𝑣.𝑠. linked to the bornologies mentioned before: the Frechet derivative linked to the bornology of the bounded subsets of a 𝑡.𝑣.𝑠., the derivative of Hadamard linked to the bornology of the compact subsets and the derivative of Gâteaux linked to the bornology of the sets finite. Our interest in this work is to develop this theory in the case of Banach spaces.

**Definition 2.1.1** Let 𝑋 and 𝑌 be Banach spaces, Ω ⊂ 𝑋 be an open subset and 𝑓: Ω → 𝑌 be a function.

We say that the function 𝑓: Ω → 𝑌 is 𝐺-differentiable (differentiable in the sense of Gâteaux) at the point 𝑎 ∊ Ω if there exists 𝑢 ∊ ℒ(𝑋, 𝑌) such that

and this limit is uniform on the finite subsets of 𝑋. In this case, we say that 𝑢 is the 𝐺-derivative of 𝑓 in 𝑎 and

We say that the function 𝑓: Ω → 𝑌 is 𝐻-differentiable (differentiable in the sense of Hadamard) at the point 𝑎 ∊ Ω if there exists 𝑢 ∊ ℒ(𝑋, 𝑌) such that

and this limit is uniform on the compact subsets of 𝑋. In this case, we say that 𝑢 is the 𝐻-derivative of 𝑓 in 𝑎 and

We say that the function 𝑓: Ω → 𝑌 is 𝐹-differentiable (differentiable in the sense of Fréchet) at the point 𝑎 ∊ Ω if there exists 𝑢 ∊ ℒ(𝑋, 𝑌) such that

and this limit is uniform on the bounded subsets of 𝑋. In this case, we say that 𝑢 is the 𝐹-derivative of 𝑓 in 𝑎 and

When differentiability holds for any 𝑎 ∊ Ω, we say that 𝑓 is 𝐺-differentiable in Ω (resp. 𝐻-differentiable) (resp. 𝐹-differentiable).

It is straightforward to see that the condition for uniform convergence is expressed as follows:

For all bounded set 𝐵 of 𝑋 and every ε > 0 there is a δ(𝐵, ε) > 0 such that

It is clear that if 𝑓 is 𝐹-differentiable in 𝑎 ∊ Ω with derivative 𝑢, then it is 𝐺-differentiable in 𝑎 with derivative 𝑢.

The following elementary theorem is fundamental to the study of differentiability of convex functions:

**Theorem 2.1.2** Let 𝑋 be a normed space. A convex continuous function defined on an open convex set with values in ℝ is necessarily locally Lipschitz.

For a proof of this theorem see the work by R. Phelps in ^{3}.

3 USCO Functions

In this section we will go deeper into the study of the class of usco functions. We start giving some basic definitions.

**Definition 3.1** (Multivalued function) Let 𝑋 and 𝑌 be topological spaces. A multivalued function of a set 𝑋 in a set 𝑌 is a correspondence 𝑥 ↦ (𝑥), which assigns to each 𝑥 ∊ 𝑋 a subset 𝑓(𝑥) of the set 𝑌. It is possible that the set (𝑥) is the empty set. The effective domain of this function is the set of the 𝑥 ∊ 𝑋 such that (𝑥) ≠ Ø.

**Definition 3.2** (Upper/Lower semicontinuous multivalued function) Let 𝑋 and 𝑌 be topological spaces. A multivalued function b: 𝑋 → 𝑌 is upper semicontinuous in 𝑎 ( 𝑋 if for every open set 𝑊 in 𝑌 such that 𝑊 ( (𝑎) (open neighborhood of 𝑓(𝑎)) there exists an open set 𝑉 of 𝑋 such that 𝑎 ∊ 𝑉 (neighborhood of 𝑎) and 𝑓(𝑥) ( 𝑊 for all 𝑥 ∊ 𝑉. In the case that 𝑓 is upper semicontinuous for any 𝑎 ∊ 𝑋, we say that 𝑓 is upper semicontinuous in 𝑋.

A multivalued function 𝑓: 𝑋 → 𝑌 is lower semicontinuous in 𝑎 ∊ 𝑋 if for every open set 𝑊 in 𝑌 such that 𝑓(𝑎) ∩ 𝑊 ≠ Ø, there exists an open set 𝑉 of 𝑋 such that 𝑎 ∊ 𝑉 (neighborhood of 𝑎) and 𝑓(𝑥) ∩ 𝑊 ≠ Ø for all 𝑎 ∊ 𝑉. In the case that 𝑓 is lower semicontinuous for any 𝑎 ∊ 𝑋, we say that 𝑓 is lower semicontinuous in 𝑋.

The first remark is that 𝑓 is continuous in 𝑎 ∊ 𝑋 if it is both upper semicontinuous and lower semicontinuous at the point 𝑎.

**Definition 3.3** (Graph of a multivalued function) Let 𝑋 and 𝑌 be topological spaces. The graph of a multivalued function 𝑓: 𝑋 → 𝑌 is the subset Graph (𝑓) of 𝑋 ( 𝑌 of the pairs (𝑥, 𝑦) ∊ 𝑋 ( 𝑌 such that 𝑦 ∊ (𝑥).

**Definition 3.4** The limit values of a net

Where ≽ is the partial order relation in 𝛵.

We note that 𝑦 is a limit value of the indicated net if and only if there exists a subnet that converges to 𝑦.

**Theorem 3.5** Let 𝑋 and 𝑌 be topological spaces. A multivalued function 𝑓: 𝑋 → 𝑌 is upper semicontinuous if and only if for any closed set 𝐶 ⊂ 𝑌, the set

is closed on 𝑋.

Proof. Suppose that 𝑓 is upper semicontinuous, we want to show that

Let 𝐶 be a closed subset of 𝑌 and 𝐴 = 𝐶^{𝐶}, then 𝐴 is an open set in 𝑌 and if (𝑥) ∩ 𝐶 = Ø, then 𝑓(𝑥) ( 𝐴. Since 𝑓 is upper semicontinuous, there is a neighborhood open 𝑉 of 𝑥 such that (𝑧) ( 𝐴 for all 𝑧 ∊ 𝑉. But this means that (𝑧) ∩ 𝐶 = Ø for all 𝑧 ( 𝑉 and, therefore,

is open set and so

is closed with 𝐶 is closed, we want to establish that 𝑓 is upper semicontinuous. Let 𝑎 ∊ 𝑋 and 𝑊 is an open subset of 𝑌 such that (𝑎) ( 𝑊. Then, 𝑊^{𝐶} is a closed set and by hypothesis

is closed in 𝑋 and its complement 𝑉 is an open set. Now, 𝑎 ∊ 𝑉 since (𝑎) ∩ 𝑊^{𝐶} = Ø and 𝑓(𝑥) ∩ 𝑊^{𝐶} = Ø for all 𝑥 ∊ 𝑉, so that 𝑓(𝑥) ⊆ 𝑊 for all 𝑋 ∊ 𝑉.

Observe that now we are able to introduce a relation of order on the set of multivalued functions (𝑋, 𝑌) with 𝑋 and 𝑌 being topological spaces. For 𝑓, 𝑔 in (𝑋, 𝑌), se define the order ≼ as

**Definition 3.6** (Usco Function) Let 𝑋 and 𝑌 be topological spaces. A multivalued function 𝑓: 𝑋 → 𝑌 is a usco function if 𝑓 is upper semicontinuous such that 𝑓(𝑥) ≠ Ø and is compact for any 𝑥 ∊ 𝑋. We denote the set of usco functions from 𝑋 to 𝑌 by 𝒰 (𝑋,𝑌).

We say that 𝑓 ∊ 𝒰 (𝑋,𝑌) is a minimal usco function if 𝑓 is a minimal element of the ordered set (𝒰 (𝑋,𝑌), ≼). This means that, if 𝑔 ∊ 𝒰 (𝑋, 𝑌) and 𝑔≼ 𝑓, then 𝑓 = 𝑔.

Now, we will establish some results about usco functions.

**Theorem 3.7** For every 𝑢 ∊ 𝒰 (𝑋,𝑌), there exists a minimal usco function 𝑓 ∊ 𝒰 (𝑋,𝑌) such that 𝑓≼ 𝑢.

Proof. Let 𝑢 ∊ 𝒰 (𝑋,𝑌) and let ℋ be the collection of usco functions ℎ such that ℎ≼𝑢. Let us show that every chain ℒ contained in ℋ is bounded below. If ℱ is a finite subset in ℒ, then

is upper semicontinuous and thus, a usco function that minorizes ℒ. By Zorn’s lemma ℋ has a minimal element.

**Lemma 3.8** Let (𝑋, 𝛼) and (𝑌, τ) be topological spaces and 𝑓: 𝑋 → 𝑌 be a usco function. If (𝑥_{𝑡},𝑦_{𝑡})_{𝑡∊𝛵} is a net in 𝐺𝑟𝑎𝑝ℎ(𝑓) and 𝑥_{𝑡} → 𝑥, then (𝑦_{𝑡})_{𝑡∊𝛵} has at least one limit value in 𝑓(𝑥).

Proof. By contradiction, we assuming that the net (𝑦_{𝑡})_{𝑡∊𝛵} has not limit values in 𝐺𝑟𝑎𝐺ℎ(𝑓). That is, for all 𝑧 ∊ 𝑓(𝑥), there exists a 𝑡_{𝑧} such that
_{𝑧} be an open neighborhood of 𝑧 such that

As 𝑥_{𝑡} → 𝑥, there exists a
_{𝑡} ∊ 𝑉 provided that
_{𝑡} ∊ 𝐴 and 𝑦_{𝑡} ∊ 𝑊𝑧 for some 𝑧 ∊ 𝐹, which implies that

**Theorem 3.9** (Characterization of usco functions) Let (𝑋, 𝛼) and (𝑌, 𝛽) be topological spaces. A multivalued function 𝑓: 𝑋 → 𝑌 is a usco function if and only if its graph 𝐺𝑟𝑎𝑝ℎ(𝑓) is a closed set. Moreover, there exists a usco function 𝑢 ∊ 𝒰 (𝑋, 𝑌) such that 𝑓 ≼ 𝑢.

Proof. Assume that 𝑓 is a usco function. If prove that
_{𝑡}, 𝑦_{𝑡})_{𝑡∊𝛵} such that 𝑥_{𝑡} → 𝑥, 𝑦_{𝑡} → 𝑦 and
_{𝑡})_{𝑡∊𝛵} has at least one adhesion value in 𝑓(𝑥). That is, there is a subnet

In the other direction, if Graph (𝑓) is closed and there is a usco function 𝑢 such that 𝑓 ≼ 𝑢, then 𝑓 is a usco function. By theorem 3.5, it suffices to show that 𝑓(𝑥) is closed for all 𝑥 ∊ 𝑋. Let
_{𝑡})_{𝑡∊𝛵} in 𝑓(𝑥) such that 𝑦_{𝑡} → 𝑦. Let 𝑥_{𝑡} = 𝑥 for all 𝑡 ∊ 𝛵. Then (𝑥_{𝑡}, 𝑦_{𝑡})_{𝑡∊𝛵} is a net in 𝐺𝑟𝑎𝑝ℎ(𝑓) that fulfills the conditions of lemma 3.8 and, therefore, a subnet (𝑦_{𝑡})_{𝑡∊𝛵} converge to 𝑦′ ∊ 𝑓(𝑥) and as 𝑦_{𝑡} → 𝑦, it follows that 𝑦′ = 𝑦 ∊ 𝑓(𝑥).

The following lemma will be helpful for the main result in this section.

**Lemma 3.10** Let (𝑋, τ) be the topological space, (𝑌, σ) Hausdorff space and 𝑓: 𝑋 → 𝑌 a usco function. The following are equivalent:

(i)𝑓 is a minimal usco function.

(ii) If 𝐴 is an open subset of 𝑋, 𝑊 is an open subset of 𝑌 and 𝑓(𝑎) ∩ 𝑊 ≠ Ø for some 𝑎 ∊ 𝐴, then there exists an open nonempty subset 𝑉 ( 𝐴 such that 𝑓(𝑥) ( 𝑊 for all 𝑥 ∊ 𝑉.

(iii)If 𝐴 is an open subset of 𝑋 and 𝐶 is a closed subset of 𝑌 such that 𝑓(𝑎) ∩ 𝐶 ≠ Ø for all 𝑎 ∊ 𝐴, then 𝑓(𝑎) ( 𝐶 for all 𝑎 ∊ 𝐴.

Proof. We will show that (𝑖) ⇒ (𝑖 𝑖). Let 𝑎 ( 𝑋 and 𝑊 ( 𝑌 open subsets as in (ii). We need to establish that exists an 𝑎_{0} ∊ 𝐴 such that 𝑓(𝑎_{0}) ( 𝑊 because the fact that 𝑓 is upper semicontinuous implies that there is an open neighborhood 𝑉 in 𝐴 of 𝑎_{0} such that

We argue by contradiction, let us assume that this statement is false and let 𝐶:= 𝑊^{𝐶} closed in 𝑌. Then, 𝑓(𝑥) ∩ 𝐶≠ Ø for any 𝑥 ∊ 𝐴. We define the function ℎ: 𝑋 → 𝑌 by

It is clear that ℎ(𝑥) ≠ Ø, closed for all 𝑥 ∊ 𝑋 and ℎ ≼ 𝑓. Then, we conclude that ℎ ∊ 𝒰 (𝑋, 𝑌) by theorem 3.9. Since 𝑓 is a minimal usco function by hypothesis, then ℎ = 𝑓 and consequently, ℎ(𝑥)= 𝑓(𝑥) ( 𝐶 for all 𝑥 ∊ 𝐴. This is a contradiction, since by hypothesis 𝑓(𝑎) ∩ 𝑊 ≠ Ø for some 𝑎 ∊ 𝐴.

Now we show that (𝑖 𝑖) ⇒ (𝑖) . By hypothesis 𝑓: 𝑋 → 𝑌 is a usco function. From theorem 3.7, there exists a minimal usco function 𝑔 ≼ 𝑓 and 𝑔 = 𝑓, as we will show below. If 𝑓 and 𝑔 are not equal, there exists 𝑥_{0} ∊ 𝑋 such that
_{0}) such that 𝑧 ∉ 𝑔(𝑥_{0}). Thus, there exists an open 𝛵 in 𝑌 such that 𝑧 ∊ 𝛵 and
_{0} such that

This construction allows us to see that 𝑥_{0} ∊ 𝐴 and 𝑓(𝑥_{0}) ∩ 𝛵 ≠ Ø. Then, there is a subset open nonempty 𝑉 ( 𝐴 such that

which is a contradiction, since

Now we see that (𝑖𝑖) ⇒ (𝑖𝑖𝑖). Let 𝐴 be a subset open in 𝑋 and 𝐶 be a subset closed in 𝑌 as in (iii). Suppose there exists 𝑎 ∊ 𝐴 such that 𝑓(𝑎) is not contained in 𝐶. Then, 𝑓(𝑎) ∩ 𝑊≠ Ø for some 𝑎 ∊ 𝐴, where 𝑊:= 𝐶^{𝐶} is an open subset in 𝑌. For (ii) there exists a nonempty open subset 𝑉 ( 𝐴 such that 𝑓(𝑥) ( 𝑊 for all 𝑥 ∊ 𝑉, i.e., 𝑓 (𝑉) ∩ 𝐶 = Ø , which is a contradiction, since by hypothesis 𝑓(𝑥) ∩ 𝐶≠ Ø for all 𝑥 ( 𝐴.

We will show that (𝑖𝑖𝑖) ⇒ (𝑖𝑖). Let 𝐴 ( 𝑋 and 𝑊 ( 𝑌, with 𝑓(𝑎_{0}) ∩ 𝑊≠ Ø such that 𝑎_{0} ∊ 𝐴. Let us prove that there exists 𝑎 ∊ 𝐴 such that 𝑓(𝑎) ( 𝑊. Otherwise, we would have 𝑓(𝑎) ∩ 𝑊^{𝐶} ≠ Ø for all 𝑎 ∊ 𝐴, where 𝐶:= 𝑊^{𝐶} is a closed subset in 𝑌. Then, for (iii) we have 𝑓(𝑎) ( 𝑊^{𝐶} for all 𝑎 ∊ 𝐴 and hence 𝑓(𝐴) ∩ 𝑊= Ø, which is a contradiction. Therefore, 𝑊 is an open neighborhood in 𝑌 of 𝑓(𝑎_{0}) and by the upper semicontinuity of 𝑓, there exists an open neighborhood

We also have the following theorem of minimal usco functions:

**Theorem 3.11** (Characterization of minimal usco functions) Let (𝑋, τ) and (𝑌,σ) be topological spaces and 𝑓: 𝑋 → 𝑌 be a usco function. Then the following statements are equivalent:

𝑓 is a minimal usco function.

For every topological space (𝑍, 𝛼) and any continuous single-valued function 𝑔 : 𝑌→ 𝑍, 𝑔 ∘ 𝑓 is a minimal usco function of 𝑋 in 𝑍.

Proof. We will show 1) ⇒ 2).

(i) 𝑔 ∘ 𝑓 is compact and nonempty for all 𝑥 ∊ 𝑋. Indeed, as 𝑓 is a usco function, then 𝑓(𝑥) is compact for all 𝑥 ∊ 𝑋. Then 𝑔(𝑓(𝑥)) is compact since 𝑔 is continuous.

(ii) 𝑔 ∘ 𝑓 is upper semicontinuous. In fact, let 𝑥_{0} ∊ 𝑋 and 𝑊 be an open neighborhood in 𝑍 such that 𝑔(𝑓(𝑥_{0})) ⊆ 𝑊. By continuity, we have 𝐴:= 𝑔^{-1} (𝑊) is open in 𝑌 such that 𝑓(𝑥_{0}) ⊆ 𝐴. Since 𝑓 is upper semicontinuous, there is a neighborhood open 𝑉 in 𝑋 of 𝑥_{0} such that 𝑓(𝑥) ⊆ 𝐴 for all 𝑥 ∊ 𝑉 and, therefore, 𝑔(𝑓(𝑥)) ⊆ 𝑊 for any 𝑥 ∊ 𝑉. So, we have shown that (𝑔 ∘𝑓) ∊ 𝒰 (𝑋, 𝑍).

(iii) It remains to show that 𝑔 ∘ 𝑓 is a minimal usco function. Using equivalences established in lemma 3.10, we must verify that 𝑔 ∘ 𝑓 satisfies (ii). Let 𝐴 be an open subset in 𝑋 and 𝑊 subset open in 𝑍 such that ((𝑔 ∘ 𝑓(𝑎)) ∩ 𝑊≠ Ø for some 𝑎 ∊ 𝐴. For continuity of 𝑔, we have 𝑔^{-1}(𝑊) is an open subset in 𝑌 such that 𝑓(𝑎) ( 𝑔^{-1} (𝑊) (this is because ((𝑔 ∘ 𝑓)(𝐴)) ∩ 𝑊≠ Ø if and only if 𝑓(𝐴) ∩ 𝑔^{-1} (𝑊) ≠ Ø. Since 𝑓 is upper semicontinuous, there is an open neighborhood 𝑉 ⊆ 𝐴 of 𝑎 such that

𝑓(𝑥) ⊆ 𝑔^{-1} (𝑊) for any 𝑥 ∊ 𝑉

and, therefore, 𝑔(𝑓(𝑥)) ⊆ 𝑊 for any 𝑥 ∊ 𝑉 .

We will show 2) ⇒ 1). Let 𝑍=𝑌 and 𝑖_{𝑌}: 𝑌→𝑌 be the identity function 𝑖_{𝑌}(𝑦)=𝑦.

We observe that 𝑖_{𝑌} ∘ 𝑓 = 𝑓, which by hypothesis is a minimal usco function.

**Definition 3.12** (Residual space) Let (𝑋, 𝛼) be a topological space. We said that 𝐺 ( 𝑋 is a residual set, if there is a family of open sets (𝒰_{𝑛})_{𝑛(ℕ} which are dense in 𝑋 such that ⋂_{𝑛∊ℕ} 𝒰_{𝑛} ⊆ 𝐺.

**Definition 3.13** (Baire space) A Baire space is a topological space (𝑋, 𝛼) with the property that any residual is dense in 𝑋.

Regarding Baire space, we have the following well known examples (see ^{4}):

1. Any locally compact topological space is a Baire space.

2. Any open subset of a Baire space is a Baire space.

3. Any complete metric space is a Baire space.

**Theorem 3.14** Let (𝑋, 𝛼) be a Baire space and (𝑌, 𝑑) be a complete metric space. Then any minimal usco function 𝑓: 𝑋 → 𝑌 is single-valued in some residual subset 𝑅 of 𝑋.

Proof. Let 𝑓: 𝑋 → 𝑌 be a minimal usco function, we will see that, for every ε > 0, there exists an open set 𝑉 such that diameter of 𝑓(𝑉) is less than ε (𝑑𝑖𝑎𝑚(𝑓(𝑉)) < ε. In fact, let 𝑦 ∊ 𝑓(𝑋) and

Let 𝐴_{ε} = ⋃{ 𝑉 ⊆ 𝑉 : 𝑋 𝑜p𝑒𝑛 and 𝑑𝑖𝑎𝑚(𝑓(𝑉)) < ε}. It is clear that this set is open. We will show that 𝐴_{ε} is dense. Let 𝑥 ∊ 𝑋 and 𝑈 be an open neighborhood of 𝑥. If

and, therefore, 𝑉 ( 𝐴_{ε}. We have shown that 𝑈 ∩ 𝐴_{ε} ≠ Ø. If ε = 𝑛^{-1}, we will write 𝐴_{𝑛} instead of
_{𝑛}: 𝑛 ∊ ℕ} is a countable collection of open subsets which are dense in 𝑋. Since this is a Baire space, then 𝑅: = ⋂ _{𝑛∊ℕ} 𝐴_{𝑛} is a subset residual dense. If 𝑥 ∊ 𝑅, then

and, therefore, 𝑓(𝑥) consists of only one element. That is, 𝑓|_{𝑅} is a single-valued function.

3.1 Usco Functions and Continuity of Subdifferential of a Convex Function

**Definition 3.1.1** (Dual space) Let X be a Banachspace. The topological dual of X is defined as the Banach space

If 𝑋 is a Banach space, the weak-* topology of the dual 𝑋′ denoted by σ (𝑋′, 𝑋) is generated by the family {𝑝_{𝑥}: 𝑥 ∊ 𝑋} of seminorms, where

We note that (𝑋′, σ (𝑋′, 𝑋)) is a completely regular topological space. A fundamental weak neighborhood of 𝑎 ∊ 𝑋 is any set of the form

Where 𝐹′ is a finite subset of 𝑋′ and ε > 0.

**Definition 3.1.2** (The subdifferential) Let Ω be a convex open subset of a Banach space 𝑋 and let 𝑓: Ω → ℝ ∪ {+∞} be a convex and continuous function, and let 𝑥 ∊ 𝑑𝑜𝑚𝑓. The subdifferential of 𝑓 in the point 𝑎 ∊ Ω, is the set of 𝑢 ∊ 𝑋′ such that

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We denote by ∂ 𝑓(𝑎) the subdifferential of 𝑓 at point 𝑎. Any function

**Definition 3.1.3** Let 𝑋 be a Banach space and Ω be a convex open subset of 𝑋. A function 𝑓: Ω → ℝ called is radially differentiable at 𝑎 ∊ Ω in the direction ℎ if there exists the limit

The value of this limit is called the radial derivative of 𝑓 at point 𝑎 in the direction ℎ. We denote

the radial derivative of 𝑓 at point 𝑎 in the direction ℎ. The function 𝑑^{+}𝑓(𝑎, ℎ) is well defined, it is sublineal, convex and continuous. If 𝑑^{+}𝑓(𝑎, ℎ) exists for all ℎ ∊ 𝑋, then we say that 𝑓 is radially differentiable at 𝑎 ∊ Ω.

We know that every convex real valued function defined in an open is radially differentiable if

then

The radial derivative 𝑑^{±}𝑓(𝑎, ℎ) it allows us to locate the subgradient at the point 𝑎 ∊ Ω in the following sense:

**Lemma 3.1.4** Let 𝑋 be a Banach space, Ω be a convex open subset of 𝑋, 𝑓: Ω → ℝ be a convex and continuous function. Then 𝑢 ∊ ∂ 𝑓(𝑎) if and only if

Proof. If 𝑢 ∊ ∂ 𝑓(𝑎), then 𝑢: 𝑋 → ℝ is linear and continuous. Also if 𝑎+ 𝑡ℎ ∊ Ω for 𝑡 small enough and

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Dividing by 𝑡 > 0 and taking 𝑡 → 0^{+}, we have 𝑢(ℎ) ≤ 𝑑^{+}𝑓(𝑎, ℎ). On the other hand,

and, therefore, when 𝑡 tends to zero by the left, we obtain that

Consequently,

and, therefore,

**Remark 3.1.5** For a Banach space 𝑋, we consider in 𝑋′ the weak-* topology σ (𝑋′, 𝑋). If 𝑓: 𝑋→ ℝ is a convex and continuous function, we will show that ∂𝑓: 𝑋 → 𝑋′ is a usco function, considering in 𝑋 the norm topology and in 𝑋′ the weak-* topology.

**Lemma 3.1.6** Let 𝑋 be a normed space, let Ω be an open and convex subset and 𝑓: Ω → ℝ be a convex and continuous function. Then,

1.The subdifferential function is a locally bounded function multivalued. In other words, for all 𝑎 ∊ Ω, exists 𝑟 > 0 and 𝑚 > 0 such that

2. If

For a proof of this lemma see the work by R Phelps ^{3}.

**Theorem 3.1.7** Let 𝑋 be a normed space, let Ω be an open and convex subset. If 𝑓: Ω → ℝ is a convex and continuous function, then 𝑥 ↦∂𝑓(𝑥) of 𝑋 in 𝑋′ is a usco function, considering in Ω the norm topology and in 𝑋′ the weak-* topology σ (𝑋′, 𝑋).

Proof. We must show that:

i. ∂𝑓(𝑥) is σ (𝑋′, 𝑋)-compact and ∂𝑓(𝑥) ≠ Ø for all 𝑥 ∊ Ω.

ii. ∂𝑓 is upper semicontinuous.

Let us show (i). Since 𝑓 is continuous at 𝑎 ∊ Ω and convex, then 𝑒𝑝𝑖(𝑓) ( 𝑋 ( ℝ is a convex set such that 𝑖𝑛𝑡(𝑒𝑝𝑖(𝑓)) ≠ Ø. In particular if (𝑎, 𝑓(𝑎)) ∉ 𝑖𝑛𝑡(𝑒𝑝𝑖(𝑓)), then under the Hahn-Banach theorem, it exists 𝑢 ∊ 𝑋′ which separates (but not strictly) the point (𝑎, 𝑓(𝑎)) of the convex set. We can observe that 𝑢 ∊ 𝑋′ is the subgradient of 𝑓 at 𝑎 and, therefore, ∂ 𝑓(𝑥) ≠ Ø for all 𝑥 ∊ Ω. Now, since ∂ 𝑓(𝑥) ( 𝑋′ is bounded, by the Banach-Alaoglu theorem, this set is σ (𝑋′, 𝑋)-relatively compact. Now, to conclude the proof we need to prove that σ (𝑋′, 𝑋)-closed. In addition, ∂ 𝑓(𝑥) is σ (𝑋′, 𝑋)-compact. We recall that, 𝑢 ∊ ∂ 𝑓(𝑥) if and only if 𝑢(𝑦) ≤ 𝑑^{+}𝑓(𝑥, 𝑦) for 𝑦 ( 𝑋 (by lemma 1.3) and

Is σ (𝑋′, 𝑋)-closed since for each 𝑦 ∊ 𝑋, the function 𝑢 ↦ 𝑢 (𝑦) is σ (𝑋′, 𝑋)-continuous.

Let us show (ii). We will argue by contradiction, assuming that 𝑥 ↦ ∂𝑓(𝑥) is not upper semicontinuous in a 𝑎 ∊ Ω. This means that there exists a σ (𝑋′, 𝑋)-neighborhood open 𝑊 of ∂𝑓(𝑎), a sequence
_{𝑛}) _{𝑛∊ℕ} has at least one σ (𝑋′, 𝑋)-adherence value 𝑢 ∊ ∂𝑓(𝑎). As 𝑊 is a σ (𝑋′, 𝑋)-neighborhood of 𝑢, then 𝑣_{𝑛} ∊ 𝑊, which is a contradiction.

**Theorem 3.1.8** Let 𝑋 be a Banach space, Ω an open and convex subset and 𝑓: Ω → ℝ is a convex and continuous function. If 𝑓 is 𝐹-differentiable in 𝑎 ∊ Ω, then the subdifferential function 𝑥↦∂𝑓(𝑥) of Ω in 𝑋′ is upper semicontinuous at 𝑥 = 𝑎 with respect to the norm topologies.

For a proof of this theorem see ^{3}.

**Lemma 3.1.9** Let w be a normed space, Ω an open and convex subset and 𝑓:Ω→ℝ a continuous convex function. A sufficient condition for that 𝑓 to be 𝐹-differentiable in 𝑎∊Ω is that there is a continuous selection in 𝑎 with respect to the norm topologies.

For a proof of this lemma see ^{3}.

4 The Asplund Spaces

We recall that a subset 𝐺 of a topological space (𝑋, 𝛼) is a set 𝐺_{δ} if it can be expressed as a countable intersection of open and dense subsets. In complete metric spaces and so, in Banach spaces, any residual subset is dense by the Baire’s theorem.

**Definition 4.1** (Asplund 𝐹-space/𝒢-space/𝒢𝒟-space) An Asplund 𝐹-space is a Banach space with the following property: If 𝑓 is a continuous convex function defined in open and convex subset 𝑈 ⊂ 𝑋, then 𝐹-differentiable in a dense subset 𝐺_{δ} of 𝑈.

A 𝒢-Asplund is a Banach space with the following property: If 𝑓 is a continuous convex function defined in open and convex subset 𝑈 ⊂ 𝑋, then 𝐺-differentiable in a dense subset 𝐺_{δ} of 𝑈.

An Asplund 𝒢𝒟-space is a Banach space with the following property: If 𝑓 is a continuous convex function defined in open subset 𝑈 ⊂ 𝑋, then 𝐺-differentiable in a dense subset 𝐺_{δ} of 𝑈.

We note that a 𝒢-Asplund space is also an Asplund 𝒢𝒟-space. We want to point out that in the usual literature, the Asplund 𝐹-spaces are called the Asplund spaces, the 𝒢-Asplund are called weak Asplund spaces and Asplund 𝒢𝒟-spaces are called weak differenciable spaces. We could talk about the 𝐻-Asplund spaces, but it does not make much sense because it has been shown that continuous convex function wich are 𝐺-differentiables are 𝐻-differentiables. In the original article by E. Asplund ^{5}, a 𝒢-Asplund is a weak differentiability spaces (𝑊𝐷𝑆) and a Asplund 𝐹-space is called strong differentiability spaces (𝑆𝐷𝑆). The definition we use comes from R. Phelps ^{6}.

We will denote by 𝔘_{𝐺} the class of the 𝒢-Asplund and 𝔘_{𝐹} the class of Asplund 𝐹-spaces.

**4.1 Asplund 𝐹-Spaces and Dentables Subset of a Banach Space**

In this section we present the proof of the Namioka-Phelps theorem, which characterizes Asplund 𝐹-spaces by means of the geometric condition of dentability. First, we discuss the notion of the dentable set. To do that, if 𝑀 is a subset of a Banach space 𝑋, we define the support function of this set 𝑀 as follows:

of 𝑋′ in ℝ ∪{∞}. If we replace 𝑀 by
_{𝑀} = 𝑝_{𝛵}. This function is sublineal and lower semicontinuous considering in 𝑋′ the weak-* topology denoted by σ (𝑋′, 𝑋) because it is the upper lower bound of lower semicontinuous functions.

Suppose now that 𝑋 is a Banach space and 𝑀′ is a subset of 𝑋′. Proceeding as before, we define the support function of the set 𝑀′ as follows:

of 𝑋′ in ℝ ∪{∞}. If
_{𝑀′} = 𝑝_{𝛵′}.

**Definition 4.1.2** (𝑋-slice) Let 𝑋 be a Banach space, 𝛼 > 0 and 𝑥 ∊ 𝑋. For a nonempty bounded subse t 𝑀 of 𝑋, we define the (𝛼, 𝑥′)-slice of 𝑀 as the set

Is clear that every slice is a relatively open set of 𝑀 with the topology σ (𝑋′, 𝑋), i.e., σ (𝑋, 𝑋′)|_{𝑀} (restriction of σ (𝑋, 𝑋′) to 𝑀).

**Definition 4.1.3** (𝑋′-slice) Let 𝑋 be a Banach space, 𝛼 > 0 and 𝑥 ∊ 𝑋. For a nonempty bounded subset 𝑀′ of 𝑋′, we define the weak*- (𝛼, 𝑥)-slice of 𝑀′ as the set

We note that ℛ′(𝛼, 𝑥; 𝑀′) is open in the topology σ (𝑋′, 𝑋)|_{𝑀′}. On the other hand, if 𝛼 < 𝛽, then

Moreover, if
_{𝑀′} = 𝑝_{𝛵′}.

**Definition 4.1.3** (Dentable set) Let 𝑋 be a Banach space. A subset 𝑀 ⊂ 𝑋 is called dentable if it admits slices of arbitrarily small diameter. We say that a subset 𝑀′ ⊂ 𝑋′ is weak *-dentable if it admits weak *-(𝛼, 𝑥)-slice of arbitrarily small diameter. A Banach space is dentable if any bounded subset is dentable.

**Theorem 4.1.4** (Namioka - Phelps). A Banach space 𝑋 is an Asplund 𝐹-space if and only if its dual 𝑋′ is weak *- dentable.

Proof. Suppose 𝑋 is an Asplund 𝐹-space and show that all bounded subset of 𝑋′ admits slices of arbitrarily small diameter, otherwise, out so there would be a bounded subset 𝑀′ of 𝑋′ in which all (𝛼, 𝑥)-slice has a diameter 𝑟 > 0.

Let

Of 𝑋 in ℝ is not 𝐹-differentiable in any point, which contradicts that 𝑋 is an Asplund 𝐹-space. With this purpose in mind, let
_{𝑋′} ≥ 𝑟 − 𝛽,

There exists a

We can now choose 𝛼 and 𝛽 appropriately. Let

Since inequality (5) holds for all 𝑡 > 0, in particular if

In virtue of theorem 2.4 with the bornology 𝛽_{𝐹}, we conclude that 𝑝 is not 𝐹-differentiable in 𝑥, which is a contradiction.

Now, we will show that 𝑋 is an Asplund 𝐹-space. Let Ω be an open subset of 𝑋 and 𝑓: Ω → ℝ a convex function. Theorem 2.5 makes possible to locate the points of differentiability of 𝑓. The bornology that we are going to consider is 𝛽_{𝐹}, of all bounded subsets 𝑋 since here the 𝐹-derivative is defined. Let 𝐷(𝑓, ε) be the set of 𝑥 ∊ Ω for which there is a δ (𝑥, ε) > 0 such that

Then:

(i) 𝐷(𝑓, ε) is open for all ε > 0.

(ii)

The proof of (ii) is obvious. We will proof (i), i.e., 𝐷(𝑓, ε) is open. Since 𝑓 is locally Lipschitzian by theorem 1.1, there exists a 𝑟 > 0 and a constant 𝑘 > 0 such that 𝐵 (𝑥, 𝑟) ( Ω and

By the definition of 𝐷(𝑓, ε), there exists δ > 0 such that

Let

Now let us observe that if 𝑡 = δ, then

Let

Once the points of 𝐹-differentiability of 𝑓 are located, it remains to show that 𝐴_{ε}: = 𝐷(𝑓, ε) is dense for every ε > 0. On the one hand, we have that Ω is a Baire space. If ε = 𝑛^{−1}, we will write
_{𝑛}: 𝑛 ∊ ℕ} is a countable collection of open subsets in Ω. Since Ω is a Baire space, if we define

(set of differentiability of 𝑓) is a set 𝐺_{δ}.

We will show that 𝐷(𝑓, ε) is dense for every ε > 0. That is, if 𝑥_{0} ( Ω (fixed but arbitrary) any open neighborhood 𝑊 of 𝑥_{0} intersects 𝐷(𝑓, ε). As the subdifferential function is locally bounded, we can suppose that ∂ 𝑓(𝑊) = 𝛵′ is a bounded subset of 𝑋′. As 𝑋′ is weak *-dentable, there exists a weak- *(𝛼, 𝑧)-slice of 𝛵′ of arbitrarily small diameter, let say < ε. If 𝑎′∊ ℛ(𝛼, 𝑧; 𝛵′), then 𝑎′∊ ∂ 𝑓(𝑎) for some 𝑎 in 𝑊, since

Since 𝑊 is open, there exists 𝑟 > 0 (sufficiently small) such that 𝑏:= 𝑎 + 𝑟𝑧 ∊ 𝑊. If 𝑏′ ∊ ∂ 𝑓(𝑏), therefore,

consequently,

As 𝑎 − 𝑏 = −𝑟𝑧 and 𝑏 − 𝑎 = 𝑟𝑧 by replacing in (6)

and how 𝑟 > 0

We have, thus, shown that

Suppose that

If

Now, as 0 < 𝑡 < δ and by the linearity of 𝑢, 𝑣, 𝑤 we have

By (7) we know that 𝑣, 𝑤 ∊ ℛ′(𝛼, 𝑧; 𝛵′) and that the diameter of this set is < ε, then we obtain

This last inequality is true for every ℎ ∊ 𝑋 such that ‖ℎ‖_{𝑋} ≤ 1 and every 0 < 𝑡 < δ. Therefore, 𝑏 ∊ 𝐷(𝑓, ε), i.e., 𝑊∩ 𝐷(𝑓, ε) ≠ Ø. This means that 𝐷(𝑓, ε) is dense and in consequence

(set of differentiability of 𝑓) is a 𝐺_{δ}-dense set. We have, thus, proved that 𝑋 is an Asplund 𝐹-space.

**5 The Class of Stegall** 𝕾 **on Topological Spaces**

This section is the central part of the article, since it introduces an intermediate class between the Asplund 𝐹-spaces and 𝒢-Asplund, called the class of Stegall

Let us recall that a topological space (𝑋, τ) is completely regular, if it is Hausdorff and for each closed set 𝐶 and every point 𝑝 that does not belong to 𝐶, there is a continuous function 𝑓: 𝑋 → [0,1] such that 𝑓|_{𝐶} = 0 and 𝑓(𝑝) = 1. The Urysohn’s lemma, every metric space is completely regular.

**Theorem 5.1** If 𝑋 is a separable Banach space and 𝐾 ⊂ 𝑋′ is bounded, then the topology σ (𝑋′, 𝑋)|_{𝐾} is metrizable (the topology of the dual is metrizable on the bounded sets).

**Theorem 5.2** Let 𝑋 be a Banach space. Then (𝑋′, σ (𝑋′, 𝑋)) is a space completely regular.

For the proof of previuos theorems see ^{1}.

**Definition 5.3** (𝑆-space) An 𝑆-space is a completely regular topological space 𝑋 that satisfies the following condition:

If 𝑍 is a Baire space and 𝑓: 𝑍 → 𝑋 is a minimal usco function, then 𝑓 is single-valued in a residual subset of 𝑍. We will denote by 𝔖, the set of all the 𝑆-spaces.

Even though Marian J. Fabian in ^{1} proves the following theorems, some important details were left out. We present here complete proofs.

**Theorem 5.4** Every metric space (𝑋, 𝑑) is an 𝑆-space.

Proof. Let (𝑋, 𝑑) be a metric space, 𝑍 be a Baire space and 𝑓: 𝑍 → 𝑋 is a minimal usco function. For each 𝑛 ∊ ℕ, we define the open set

We will show that
_{𝑛} is open and we will prove that each 𝒰_{𝑛} is dense for all 𝑛 ∊ ℕ. Let 𝑥 ∊ 𝒰_{𝑛} and 𝑉_{𝑥} ⊂ 𝑍 be an open neighborhood of 𝑥 with

If
_{𝑥} such that

Then, Ω ⊂ 𝒰_{𝑛} and, therefore, Ø ≠ 𝑉_{𝑥} ∩ 𝒰_{𝑛}, which proof the density of the 𝒰_{𝑛}. So
_{𝑛} is a residual subset. Finally, by vertue of theorem 3.14, the usco function 𝑓 is single-valued in 𝐺 and, consequently, 𝑋 ( 𝔖.

**Theorem 5.5** Let 𝑋 be a completely regular space. If
_{𝑛} , with 𝑋 ( 𝔖 for every 𝑛 ∊ ℕ and 𝑋=⋃_{𝑛∊ℕ} 𝑋_{𝑛}, then 𝑋 ∊ 𝔖.

Proof. Let 𝑍 be a Baire space and 𝑓: 𝑍 → 𝑋 a minimal usco function. For each 𝑛 ∊ ℕ, we define the set

Since 𝑓 is a upper semicontinuous function and by hypothesis each 𝑋_{𝑛} is closed in 𝑋, then each 𝑍_{𝑛} is a closed set in 𝑍 by theorem 3.5. If
_{𝑛∊ℕ} 𝑍_{𝑛} (see ^{7}, p.63). It is clear that

𝐴_{𝑛} ≠ for some 𝑛 ∊ ℕ

Let 𝑛 ∊ ℕ be such that 𝐴_{𝑛} ≠ Ø, and let
_{𝑛} is a minimal usco function and 𝑓_{𝑛}(𝐴_{𝑛}) ( 𝑋_{𝑛}.

1) In general, if 𝑓: 𝑍 → 𝑋 is a usco function and 𝑔: 𝑋 → 𝑌 (with 𝑌 a regular space) is a continuous single-valued function, by theorem 3.11.2, we have that 𝑔 ∘ 𝑓 is a usco function. From this it follows that if 𝑋 is a subspace of 𝑌, then 𝑓|_{𝑋} is a usco function since
_{𝑋} is continuous.

2) We will show that 𝑓_{𝑛} is minimal. For this we will use lemma 3.10 several times. Let

3)

Then

and as 𝑓 is minimal, by lemma 3.10.(iii) then 𝑓(𝑧) ( 𝐶 for every 𝑧 ∊ 𝑉. So

and, hence, 𝑓_{𝑛} is minimal (using lemma 3.10 again).

1) 𝑓_{𝑛}(𝐴_{𝑛}) ( 𝑋_{𝑛}. By the definition 𝑍_{𝑛}, we have
_{𝑛} and by lemma 3.10.(iii), we obtain that
_{𝑛} is closed by hypothesis.

Let

𝑅: = {𝑧 ∊ 𝑍: 𝑓(𝑧) is a singleton 𝑓 or all 𝑧}

And 𝑅_{𝑛}: = 𝑅 ∩ 𝐴_{𝑛}.

We note that 𝑅_{𝑛} is the set of 𝑧 ∊ 𝐴_{𝑛} such that
_{𝑛} ∊ 𝔖, so 𝐴_{𝑛} is a Baire space because it is an open subset of a Baire space and 𝑓_{𝑛}: 𝐴_{𝑛}→ 𝑋_{𝑛} is a minimal usco function. Then we conclude that 𝑅_{𝑛} is a residual set in 𝐴_{𝑛} for each 𝑛∊ℕ. Now we can show that 𝑅 is residual in 𝑍. We have shown that there exists a countable collection
_{𝑛} is residual in 𝐴_{𝑛} and

Then
_{𝑛} ( 𝐴_{𝑛} for each 𝑛∊ℕ. Let us show that 𝑊 = ⋃_{𝑛∊ℕ} 𝑊_{𝑛} is dense in 𝑍. Let 𝑧 ∊ 𝑍 and 𝑉 be a open neighborhood of 𝑧 contained in 𝑍. As 𝑉 ∩ 𝐴 ≠ Ø, then 𝑉 ∩ 𝐴_{𝑛} ≠ Ø for some 𝑛∊ℕ. Let 𝑝 be the first ordinal such that 𝑉 ∩ 𝐴_{𝑝} ≠ Ø. Then 𝑉 ∩ 𝑊_{𝑝} ≠ Ø and, therefore, 𝑉 ∩ 𝑊 = 𝑉 ∩ ⋃_{𝑛∊ℕ} 𝑊_{𝑛} ≠ Ø which shows that 𝑊 is dense at 𝑍.

Since 𝑅 ∩ 𝐴_{𝑛} is residual in 𝐴_{𝑛}, there exist a countable collection
_{𝑛} such that

The set
_{𝑛}. In fact, let 𝑧 ( 𝑊_{𝑛} and 𝑉 be a open neighborhood of 𝑧 contained in 𝑊_{𝑛}. Then 𝑉 is an open neighborhood of 𝑧 contained in 𝐴_{𝑛}. Since 𝛵_{𝑘𝑛} is dense in 𝐴_{𝑛}, then
_{𝑛} is a Baire space, then

This is the final part of the proof. Let
_{𝑘} is open and dense in 𝑍. To prove this statement, let 𝑉 be an open and nonempty subset of 𝑍. Since 𝑊 is dense in 𝑍, then
_{𝑛}). As
_{𝑛}, it follows that

We have, thus, shown that
_{𝑘} is open and dense in 𝑍. As 𝑍 is a Baire space, then

As

Since

Using (9), we obtain that

In other words, we have shown that 𝑅 is residual, since each 𝛵_{𝑘} is open and dense, accordingly 𝑋 = ⋃_{𝑛∊ℕ} 𝑋_{𝑛} ∊ 𝔖.

5.1 The Class of Stegall

**of Banach Spaces**

The class

Next, we give a detailed proof of the Stegall’s theorem whose proof can be found in ^{1} citing numerous articles. We show that the usco multivalued functions, particulary the subdifferentials of convex functions, play an important role in this proof.

**Theorem 5.1.1** The Stegall class lies between the Asplund 𝐹-spaces and the 𝒢-Asplund. In other words,

Proof. Let us first show that

A weak-*(𝛼, 𝑥)-slice of 𝑀 is a set σ (𝑋′, 𝑋)|_{𝑀} open due to the continuity of the function 𝑥′ ↦ ⟨ 𝑥, 𝑥′⟩ (i.e., the continuity of the support function).

Suppose 𝑋 is an Asplund 𝐹-space and let
_{𝑀} (restriction of σ (𝑋′, 𝑋) to 𝑀) and let 𝛵 be a minimal usco function. We will show that 𝛵 is single-valued in a residual subset. This would show that 𝑀 is an 𝑆-space and since 𝑀 is σ (𝑋′, 𝑋)-closed and

(since 𝑋′ is fitted with the topology
_{𝑋}, ∊ τ(0) we conclude that 𝑋′ is an 𝑆-space by theorem 5.5, that is,

Our aim now is to show that 𝛵 is single-valued in a residual subset. For this, let us consider the set defined in (8).

We claim that 𝒰_{𝑛} is an open set in (𝑀, τ), where τ = σ (𝑋′, 𝑋)|_{𝑀}. In fact, If 𝑤 ∊ 𝑉, then 𝑉 is an open neighborhood of 𝑤.

Now, we claim that 𝒰_{𝑛} is dense in 𝑍. To prove this statement, let 𝒰 be an open and nonempty subset of 𝑍. Recall that 𝑋 is an Asplund space, so from theorem 1.1, 𝑋′ is weak-*-dentable. Now, since 𝛵(𝒰) ( 𝑀, then given ε > 0 there exists a weak-*(𝛼, 𝑥)-slice ℛ(𝛼, 𝑥; 𝛵(𝒰)) of arbitrarily small diameter, let say < ε (i.e., 𝑑𝑖𝑎𝑚[ ℛ(𝛼, 𝑥; 𝛵(𝒰))] < ε). If

then 𝑊 is a subset τ-open of 𝑀 and

By lemma 3.10.(ii), there exists an open nonempty subset 𝑆 ( 𝒰 such that 𝛵(𝑆) ( 𝑊 and, therefore, 𝛵(𝑆) ( 𝛵(𝒰) and 𝛵(𝑆) ( 𝑊 ∩ 𝛵(𝒰) = ℛ(𝛼, 𝑥; 𝛵(𝒰)). So that,

which shows that 𝒰_{𝑛} is dense in 𝑍. From previous claims, we have that {𝒰_{𝑛}: 𝑛∊ℕ} is a countable collection of open and dense subsets in 𝑍. We remark that

Indeed, if

that is, 𝛵(𝑧) is a singleton for all 𝑧 from the dense residual subset 𝒟 of 𝑍. We have, thus, shown that 𝛵 is single-valued in a residual subset of 𝑍, and so

Now, we will establish that

By theorem 3.1.5, the subdifferential ∂ ψ is a usco function considering in Ω the norm topology and in 𝑋′ the weak-* topology σ (𝑋′, 𝑋). By theorem 3.7, there exists a minimal usco function 𝛵: Ω → 𝑋′ such that 𝛵 ≼ ∂ ψ. Since Ω is a Baire space, (𝑋′, σ (𝑋′, 𝑋)) is completely regular and 𝛵: Ω → 𝑋′ is a minimal usco function, then 𝛵 is single-valued on a residual subset 𝒟 of Ω. It remains to verify that ψ is 𝐺-differentiable in 𝒟. Let 𝑥 ∊ 𝒟 (arbitrary but fixed) and ℎ ∊ 𝑋 . Then 𝛵(𝑥) = {𝑢}. As }.(𝑥) ( ∂ ψ (𝑥), then 𝑢 ∊ ∂ ψ (𝑥) and, therefore, for a 𝑡 > 0 small enough we have to

Let

which implies that

Since 𝑡 > 0, from (10) and (11), we obtain that

and, therefore,

Now, let ε > 0 be such that

is a σ (𝑋′, 𝑋)-open neighborhood 𝑢 ∊ 𝑋′. Using the upper semicontinuity of 𝛵 in 𝑥 ∊ Ω, there exists an open neighborhood 𝑉 ( Ω of 𝑥 such that

There exists a δ > 0 such that

In virtue of (12) and (13), we conclude that

Since 𝑢 ∊ 𝑋′, so

We have, thus, shown that ψ is 𝐺-differentiable in a 𝐺_{δ}-dense subset 𝒟 and hence 𝑋∊ 𝔘_{𝐺}.

**Proposition 5.1.2** Every separable Banach space belongs to the Stegall class.

Proof. Suppose 𝑋 is a separable Banach space. Let

By the Banach-Alaoglu theorem, the closed unitary ball 𝐵′ de 𝑋′ is σ (𝑋′, 𝑋)-compact, then by theorem 5.1 we have that (𝐵_{𝑋′}, τ) is metrizable and (𝐵_{𝑋′}, τ) ∊ 𝔖. As

Then (𝑋′, σ (𝑋′, 𝑋)) ∊ 𝔖 by theorem 5.5 and therefore

**Remarks 5.1.3**

1) In general
^{1}(ℝ). It has been showed that the ‖·‖_{1} is not 𝐹-differentiable at any point, but by theorem 5.1.2 we deduce that 𝓁^{1}(ℝ) is a Stegall space because it is separable. Consequently, 𝓁^{1}(ℝ) ∊ 𝔖, but 𝓁^{1}(ℝ) ∉ 𝔘_{𝐹}.

2) In general
^{8}). Kalenda in ^{8} showed that there are 𝒢-Asplund spaces whose dual with the weak weak-* topology do not belong to class

3) There exist Banach spaces that are not 𝒢-Asplund spaces, for example, the nonseparable Banach space 𝑋 = 𝓁^{∞}( ℝ) is not a 𝒢-Asplund space.

5.2 Example of Usco Function

**The Supremum Mapping**

As it is well known, the problem of the differentiability of convex functions is well addressed by Fabian in ^{1}. Let 𝑋 be a compact and Hausdorff topological space, and 𝒞(𝑋) be the Banach space of the continuous functions of 𝑋 in ℝ with the norm

φ: 𝒞(𝑋) → ℝ

This function is sublineal and, therefore, convex. In addition, we define

ψ: 𝒞(𝑋) → 𝑋

This is a multivalued function that assigns to each function 𝑓 the set of points 𝑡 ∊ 𝑋 in which the function attains the supremum. We will call ψ the supremum mapping. We remark that the function 𝑓 ↦ ψ(𝑓) is a usco function. Indeed,

I. It is clear that ψ(𝑓) is compact because

is closed in 𝑋. As 𝑋 is compact,

and, therefore, ψ(𝑓) ≠ Ø for all 𝑓 ( 𝒞(𝑋).

II. We will show that ψ is upper semicontinuous. We argue by contradiction assuming that for some point 𝑔 ∊ 𝒞(𝑋) there is an open neighborhood ψ(𝑔) ( 𝑊 such that on every open ball 𝐵(𝑔, ε) there exists a ℎ such that ψ(ℎ) is not contained in 𝑊. That is, there exists a sequence (𝑔_{𝑛})_{𝑛∊ℕ} such that

As 𝑊^{𝐶} is compact, the sequence (𝑡)_{𝑛∊ℕ} has limit value 𝑡 ∊ 𝑊^{𝐶}. Since 𝑔 is a continuous function, 𝑔(𝑡) is an adhesion value of the sequence (𝑔(𝑡)) _{𝑛∊ℕ} in ℝ. Therefore, there is a subsequence

In summary, ψ is a usco function. We emphasize that this example is closely related to the differentiability of the supremum norm in 𝒞(𝑋).