International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2021 / Article

Research Article | Open Access

Volume 2021 |Article ID 6320969 | https://doi.org/10.1155/2021/6320969

Oussama Kabbouch, Mustapha Najmeddine, "The Reverse of the Intermediate Value Theorem in Some Topological Spaces", International Journal of Mathematics and Mathematical Sciences, vol. 2021, Article ID 6320969, 4 pages, 2021. https://doi.org/10.1155/2021/6320969

The Reverse of the Intermediate Value Theorem in Some Topological Spaces

Academic Editor: Frédéric Mynard
Received04 May 2020
Revised07 Mar 2021
Accepted23 Mar 2021
Published12 Apr 2021

Abstract

Any continuous function with values in a Hausdorff topological space has a closed graph and satisfies the property of intermediate value. However, the reverse implications are false, in general. In this article, we treat additional conditions on the function, and its graph for the reverse to be true.

1. Introduction

The 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)The intermediate value property (IVP).(2)Its graph is closed.(3)The inverse image of each closed set is closed.

The reverse of the two first results is false in general. That means if a function satisfies the IVP or its graph is closed, it is not necessarily continuous. The objective of this work is to provide some conditions in which this reverse is true. In this sense, we prove in Theorem 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. Theorem 1 is a generalization of Theorem 2 for functions having a closed graph from locally connected space into locally compact spaces. In Theorem 1, we have a generalization in normed spaces of the result given in [2]. In Theorem 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]. The 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 and be Hausdorff topological spaces and be a map from into . The graph of is defined by

Definition 1. Let be a Hausdorff topological space.(1)The function is said to be satisfying the intermediate value property (IVP) if the image of every connected subset of is connected in .(2)A subset of is said to be sequentially closed if it contains the limit of every convergent sequence in .(3)A subset of is said to be sequentially compact if every sequence in admits a limit point in .(4)The space is said to be locally sequentially compact if every admits a basis of sequentially compact neighbourhoods.(5)The space is said to be locally connected if every 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 is continuous, then its graph is closed and it satisfies the IVP.

Theorem 1. Let be a connected subset in a topological space . For every subset of such that and , we have , where .

Now, consider the following examples.

Example 1. (1)The functionsatisfies the IVP, but it is not continuous at 0.(2)The functionhas 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 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 be an uncountable set and be the real or the complex field. Set with andThen, is a linear subspace of the space of bounded sequences in . Define a duality between end for all and byThe setswhere is a finite subset of and , form a basis of a zero neighbourhood of a linear topology on called the weak topology on with respect to the duality between and denoted by .
The sets where is a convex balanced compact subset in the weak topology of and form a basis of a zero neighbourhood of a linear topology on called the Mackey topology on with respect to the duality between and denoted by .
The Mackey topology on is the strongest topology for which the topological dual is the space of the linear mapsLet be provided with the Mackey topology and be the Banach space of summable families . The canonical injection from into 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 is a locally connected metric space and is locally sequentially compact. If satisfies the IVP and its graph is sequentially closed, then is continuous.

Proof. Suppose that is not continuous at . Since is locally connected, there exists a connected neighbourhood of such thatSet ; then, is neighbourhood of . Therefore, there exists a connected neighbourhood of such that , and so we construct a decreasing sequence of connected neighbourhoods of such thatOn the other hand, since is not continuous at and is locally sequentially compact, there exists a sequentially compact neighbourhood of such thatThen,Since is connected for all , by Theorem 1, we haveHence, for all , there exists such that . With the sequential compactness of , there is a subsequence of which converges to . Since the sequence is in and converges to , then and , which is a contradiction because is a neighbourhood of . Then, is continuous at .

Remark 2. By the previous theorem,(1)The graph of the function given in Example 1 is not a closed set.(2)The function , given in Example 2, does not satisfy the IVP.

Theorem 3. Let be a locally connected topological space and be locally compact. If satisfies the IVP and its graph is closed, then is continuous.

Proof. Suppose that is not continuous at . Since is locally compact, there is a compact neighbourhood of such that for all neighbourhood of . By the local connectedness of , there is a generalized sequence of connected neighbourhoods of , which is a basis of neighbourhoods of in . Then,Since satisfies the IVP, is connected for all . By Theorem 1, there exists a generalized sequence such thatBy the compactness of , is compact. Then, there exists a subsequence such that converges to . Then, the generalized sequence converges to . Since is closed, then . Therefore, and , which contradicts that is a neighbourhood of . Thus, 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.

Theorem 4. Let be a real valued function on an interval of satisfying the IVP. If for all is closed in , then is continuous.

Proposition 2. Let be a function; then:(1)If the graph of is closed, then for every is closed in .(2)If the graph of is sequentially closed, then for every is sequentially closed in .(3)If is continuous, then for every is closed in .

Proof. (1)Let be a generalized sequence in that converges to . is a sequence in that converges to . Since is closed, then . Hence, . Therefore, is closed in .(2)In the same way, we show (2).(3)The continuity of implies that the graph of is closed, and then we have (3).

Remark 3. (1)The reverse of 1 and 2 is false, in general. As in Example 2, since is injective, is closed in , but has no closed graph.(2)Theorem 2 is a corollary of Theorem 4 and Proposition 2.

The following theorem is a generalization of the real case in Theorem 4.

Theorem 5. Suppose that is a normed vector space over or and is locally connected metric space. If satisfies the IVP and the inverse image of every sphere in is sequentially closed in , then is continuous.

Proof. Suppose that is not continuous at . As in the proof of Theorem 2, there is and a sequence in that converges to , and for all , where are the ball and the sphere of radius , respectively. Hence, is in which is closed. Then, and : a contradiction. Thus, is continuous.

Theorem 6. Suppose that is a finite dimensional normed vector space. If the graph of is sequentially closed, then the inverse image of every sphere in is sequentially closed in .

Proof. Let be the sphere of center and of radius . Let be a sequence in that converges to . Let us show that . We know that the sequence is in . Since is a finite dimensional normed space, then is compact. Hence, there is a subsequence of that converges to . Since the graph of is sequentially closed and is a sequence in that converges to , then in and . Hence, and . Therefore, is sequentially closed in .

Theorem 7. Suppose that is a finite dimensional normed space. If the graph of is closed, then the inverse image of every sphere in is closed in .

Proof. Let be the sphere of center and of radius . Let be in the closure ; then, there exists a generalized sequence in that converges to . Then, the sequence is in which is compact. Hence, there is a subsequence that converges to . The graph of is closed and is in ; then, . Hence, and . Therefore, is closed in .

Corollary 2. Suppose that is a finite dimensional normed space over or and is locally connected metric space. If satisfies the IVP and its graph is sequentially closed, then is continuous.

Data Availability

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

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

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  2. W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, New York, NY, USA, 1964.
  3. L. Schwartz, Analyse: Topologie Génerale Et Analyse Fonctionnelle, Hermann, Paris, France, 1970.
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Copyright © 2021 Oussama Kabbouch and Mustapha Najmeddine. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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