The Reverse of the Intermediate Value Theorem in Some Topological Spaces

+e reverse of the two first results is false in general. +at means if a function satisfies the IVP or its graph is closed, it is not necessarily continuous. +e objective of this work is to provide some conditions in which this reverse is true. In this sense, we prove in +eorem 2 that every function from a locally connected metric space into a locally sequentially compact space, which has a sequentially closed graph and satisfies the IVP, is continuous.+eorem 1 is a generalization of +eorem 2 for functions having a closed graph from locally connected space into locally compact spaces. In +eorem 1, we have a generalization in normed spaces of the result given in [2]. In+eorem 6, we show that the sequential closeness of the graph implies that the inverse image of a sphere is sequentially closed. We know that the closeness of the graph implies the continuity of linear maps between Banach spaces [3]. +e same result is given in Corollary 2 for functions, not necessarily linear, between normed spaces satisfying the IVP in the case if the codomain is a finite dimensional normed space. From now, Let E and F be Hausdorff topological spaces and f be a map from E into F. +e graph of f is defined by


Introduction
e motivation of this work derives from the intermediate value theorem (IVT) and the closed graph theorem [1]. We discuss the results presented in [2], showing that the reverse of the IVT may be true under certain additional assumptions.
A continuous function in a Hausdorff space is known to satisfy the following facts: (1) e intermediate value property (IVP).
(3) e inverse image of each closed set is closed. e reverse of the two first results is false in general. at means if a function satisfies the IVP or its graph is closed, it is not necessarily continuous. e objective of this work is to provide some conditions in which this reverse is true. In this sense, we prove in eorem 2 that every function from a locally connected metric space into a locally sequentially compact space, which has a sequentially closed graph and satisfies the IVP, is continuous. eorem 1 is a generalization of eorem 2 for functions having a closed graph from locally connected space into locally compact spaces. In eorem 1, we have a generalization in normed spaces of the result given in [2]. In eorem 6, we show that the sequential closeness of the graph implies that the inverse image of a sphere is sequentially closed.
We know that the closeness of the graph implies the continuity of linear maps between Banach spaces [3]. e same result is given in Corollary 2 for functions, not necessarily linear, between normed spaces satisfying the IVP in the case if the codomain is a finite dimensional normed space.
From now, Let E and F be Hausdorff topological spaces and f be a map from E into F. e graph of f is defined by (1) Definition 1. Let X be a Hausdorff topological space.
(1) e function f is said to be satisfying the intermediate value property (IVP) if the image of every connected subset of E is connected in F. (2) A subset A of X is said to be sequentially closed if it contains the limit of every convergent sequence (a n ) n∈N in A. (3) A subset A of X is said to be sequentially compact if every sequence in A admits a limit point in A. (4) e space X is said to be locally sequentially compact if every x ∈ X admits a basis of sequentially compact neighbourhoods. (5) e space X is said to be locally connected if every x ∈ X admits a basis of connected neighbourhoods.

Remark 1.
Every sequentially closed subspace of a locally sequentially compact space is locally sequentially compact.
In [1], we have the following classical results.
Proposition 1. If f is continuous, then its graph is closed and it satisfies the IVP.
Now, consider the following examples.
satisfies the IVP, but it is not continuous at 0.
has a closed graph, but it is not continuous at 0. (3) By the Darboux theorem [4] (asserting that the derivative of a differentiable function on the real line satisfies the IVP), the derivative of any real function which is not in C 1 on the real line satisfies the IVP but it is not continuous. (4) Using the expansion of reals in basis 13, the work given in [5] gives a construction of functions satisfying the IVP which are nowhere continuous. (5) Let I be an uncountable set and K be the real or the complex field.
en, E is a linear subspace of the space ℓ ∞ (I) of bounded sequences in K I . Define a duality between E end F for all x � (x i ) i∈I ∈ E and y � ( e sets where A is a finite subset of E and ε > 0, form a basis of a zero neighbourhood of a linear topology on F called the weak topology on F with respect to the duality between E and F denoted by σ(F, E). e sets W(K, ε) where K is a convex balanced compact subset in the weak topology of F and ε > 0 form a basis of a zero neighbourhood of a linear topology on E called the Mackey topology on E with respect to the duality between E and F denoted by τ(E, F).
e Mackey topology on E is the strongest topology for which the topological dual is the space of the linear maps Let E be provided with the Mackey topology τ(E, F) and ℓ 1 (I) be the Banach space of summable families (x i ) i∈I . e canonical injection from E into ℓ 1 (I) is a map with a closed graph that is not continuous. For more details, see [6].
In the following, we study conditions making the reverse of IVT true.

Theorem 2. Suppose that (E, d) is a locally connected metric space and F is locally sequentially compact. If f satisfies the IVP and its graph is sequentially closed, then f is continuous.
Proof. Suppose that f is not continuous at a ∈ E. Since E is locally connected, there exists a connected neighbourhood C 0 of a such that Set U � C 0 ∩ B(a, (1/2)); then, U is neighbourhood of a. erefore, there exists a connected neighbourhood C 1 of a such that C 1 ⊂ U, and so we construct a decreasing sequence (C n ) n∈N of connected neighbourhoods of a such that C n ⊂ B a, 1 2 n , for all n ∈ N.
On the other hand, since f is not continuous at a and F is locally sequentially compact, there exists a sequentially compact neighbourhood V of f(a) such that f C n ⊄V, for all n ∈ N. (10) en, (11) Since f(C n ) is connected for all n ∈ N, by eorem 1, we have f C n ∩ Fr(V) ≠ ∅, for all n ∈ N. (12) Hence, for all n ∈ N, there exists a n ∈ C n such that f(a n ) ∈ Fr(V). With the sequential compactness of Fr(V), there is a subsequence (f(a h(n) )) n∈N of (f(a n )) n∈N which converges to b ∈ Fr(V). Since the sequence 2 International Journal of Mathematics and Mathematical Sciences

(a h(n) , f(a h(n) )) n∈N is in Γ f and converges to (a, b), then b � f(a) and f(a) ∈ Fr(V), which is a contradiction because V
is a neighbourhood of f(a). en, f is continuous at a. □ Remark 2. By the previous theorem, (1) e graph of the function f given in Example 1 is not a closed set. (2) e function g, given in Example 2, does not satisfy the IVP.

Theorem 3. Let E be a locally connected topological space and F be locally compact. If f satisfies the IVP and its graph is closed, then f is continuous.
Proof. Suppose that f is not continuous at a ∈ E. Since F is locally compact, there is a compact neighbourhood V of f(a) such that f(U) ⊄V for all neighbourhood U of a. By the local connectedness of E, there is a generalized sequence (U α ) α∈Λ of connected neighbourhoods of a, which is a basis of neighbourhoods of a in E. en, (13) Since f satisfies the IVP, f(U α ) is connected for all α ∈ Λ. By eorem 1, there exists a generalized sequence (x α ) α∈Λ such that (14) By the compactness of V, Fr(V) is compact. en, there exists a subsequence (x α ) α∈Δ ⊂ Λ such that (f(x α )) α∈Δ converges to b ∈ Fr(V). en, the generalized sequence (x α , f(x α )) α∈Δ converges to (a, b). Since Γ f is closed, then (a, b) ∈ Γ f . erefore, b � f(a) and f(a) ∈ Fr(V), which contradicts that V is a neighbourhood of f(a). us, f is continuous.

□
Since the normed spaces are locally connected and the finitely dimensional normed spaces are locally compact, we have the following corollary.

Corollary 1. Each function from a normed vector space into a finitely dimensional normed vector space which satisfies the IVP and with closed graph is continuous.
In [2], the following theorem gives the reverse of the IVT under weak assumptions. (1) If the graph of f is closed, then for every y ∈ F, f − 1 ( y ) is closed in E. (2) If the graph of f is sequentially closed, then for every (

3) If f is continuous, then for every
(2) In the same way, we show (2).
(3) e continuity of f implies that the graph of f is closed, and then we have (3).

Remark 3
(1) e reverse of 1 and 2 is false, in general. As in Example 2, since f is injective, f − 1 ( y ) is closed in R, but f has no closed graph. (2) eorem 2 is a corollary of eorem 4 and Proposition 2.
e following theorem is a generalization of the real case in eorem 4.

Theorem 5.
Suppose that F is a normed vector space over R or C and E is locally connected metric space. If f satisfies the IVP and the inverse image of every sphere in F is sequentially closed in E, then f is continuous.
Proof. Suppose that f is not continuous at a ∈ E. As in the proof of eorem 2, there is r > 0 and a sequence (a n ) n∈N in E that converges to a, and for all n ∈ N, f(a n ) ∈ Fr(V) � S, where V � B(f(a), r) and S(f(a), r)are the ball and the sphere of radius f(a), respectively. Hence, (a n ) n∈N is in f − 1 (S) which is closed. en, a ∈ f − 1 (S) and f(a) ∈ S: a contradiction. us, f is continuous. □ Theorem 6. Suppose that F is a finite dimensional normed vector space. If the graph of f is sequentially closed, then the inverse image of every sphere in F is sequentially closed in E.
} be the sphere of center a and of radius r > 0. Let (x n ) n∈N be a sequence in f − 1 (S) that converges to x. Let us show that x ∈ f − 1 (S). We know that the sequence (f(x n )) n∈N is in S. Since F is a finite dimensional normed space, then S is compact. Hence, there is a subsequence (f(x h(n) )) n∈N of (f(x n )) n∈N that converges to y ∈ S. Since the graph Γ f of f is sequentially closed and (x h(n) , f(x h(n) )) n∈N is a sequence in Γ f that converges to (x, y), then (x, y) in Γ f and y � f(x). Hence, f(x) ∈ S and x ∈ f − 1 (S). erefore, f − 1 (S) is sequentially closed in E. □ Theorem 7. Suppose that F is a finite dimensional normed space. If the graph of f is closed, then the inverse image of every sphere in F is closed in E.

International Journal of Mathematics and Mathematical Sciences
Proof. Let S � S(a, r) � x ∈ F: ‖x − a‖ � r { } be the sphere of center a and of radius r > 0. Let x be in the closure f − 1 (S); then, there exists a generalized sequence (x α ) α∈Λ in f − 1 (S) that converges to x. en, the sequence (f(x α )) α∈Λ is in S which is compact. Hence, there is a subsequence (f(x α )) α∈Δ ⊂ Λ that converges to y ∈ S.
e graph Γ f of f is closed and (x α , f(x α )) α∈Δ is in Γ f ; then, y � f(x). Hence, f(x) ∈ S and x ∈ f − 1 (S). erefore, f − 1 (S) is closed in E. □ Corollary 2. Suppose that F is a finite dimensional normed space over R or C and E is locally connected metric space. If f satisfies the IVP and its graph is sequentially closed, then f is continuous.

Data Availability
e data used to support the findings of this study have not been made available because they are confidential.

Conflicts of Interest
e authors declare that they have no conflicts of interest.