# The Reverse of the Intermediate Value Theorem in Some Topological Spaces

Authors: Oussama Kabbouch; Mustapha Najmeddine
Journal: International Journal of Mathematics and Mathematical Sciences
 (2021)

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## 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.


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## Body

## 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 Theorem2 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, LetE and F be Hausdorff topological spaces and f be a map from E into F. The graph of f is defined by(1)Γf=x,fx:x∈E.Definition 1.
LetX be a Hausdorff topological space.(1)
The functionf 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 subsetA of X is said to be sequentially closed if it contains the limit of every convergent sequence ann∈ℕ in A.(3)
A subsetA of X is said to be sequentially compact if every sequence in A admits a limit point in A.(4)
The spaceX is said to be locally sequentially compact if every x∈X admits a basis of sequentially compact neighbourhoods.(5)
The spaceX 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.
Iff is continuous, then its graph is closed and it satisfies the IVP.Theorem 1.
LetC be a connected subset in a topological space X. For every subset A of X such that C∩A≠∅ and C∩X∖A≠∅, we have C∩FrA≠∅, where FrA=A¯∖A°.Now, consider the following examples.Example 1.
(1)
The function(2)fx=sin1x,ifx≠0,0,ifx=0satisfies the IVP, but it is not continuous at 0.(2)
The function(3)gx=1x,ifx≠0,0,ifx=0has 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 C1 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)
LetI be an uncountable set and K be the real or the complex field. Set E=KI=⊕i∈IKi with Ki=K,i∈I and(4)F=xii∈I∈KI:i∈I:xi≠0is at most countable.
Then,E is a linear subspace of the space ℓ∞I of bounded sequences in KI. Define a duality between E end F for all x=xii∈I∈E and y=yii∈I∈F by(5)x,y=∑i∈Ixiyi.
The sets(6)WA,ε=y∈F:x,y<εfor allx∈A,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 σF,E.
The setsWK,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 τE,F.
The Mackey topology onE is the strongest topology for which the topological dual is the space of the linear maps(7)φy:E⟶ℝ,x↦x,y,y∈F.
LetE be provided with the Mackey topology τE,F and ℓ1I be the Banach space of summable families xii∈I. The canonical injection from E into ℓ1I 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 thatE,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 thatf is not continuous at a∈E. Since E is locally connected, there exists a connected neighbourhood C0 of a such that(8)C0⊂Ba,1=x∈E,da,x<1.
SetU=C0∩Ba,1/2; then, U is neighbourhood of a. Therefore, there exists a connected neighbourhood C1 of a such that C1⊂U, and so we construct a decreasing sequence Cnn∈ℕ of connected neighbourhoods of a such that(9)Cn⊂Ba,12n,for alln∈ℕ.
On the other hand, sincef is not continuous at a and F is locally sequentially compact, there exists a sequentially compact neighbourhood V of fa such that(10)fCn⊄V,for alln∈ℕ.
Then,(11)fa∈fCn∩V,fCn∩F∖V≠∅,for alln∈ℕ.
SincefCn is connected for all n∈ℕ, by Theorem 1, we have(12)fCn∩FrV≠∅,for alln∈ℕ.
Hence, for alln∈ℕ, there exists an∈Cn such that fan∈FrV. With the sequential compactness of FrV, there is a subsequence fahnn∈ℕ of fann∈ℕ which converges to b∈FrV. Since the sequence ahn,fahnn∈ℕ is in Γf and converges to a,b, then b=fa and fa∈FrV, which is a contradiction because V is a neighbourhood of fa. Then, f is continuous at a.Remark 2.
By the previous theorem,(1)
The graph of the functionf given in Example 1 is not a closed set.(2)
The functiong, given in Example 2, does not satisfy the IVP.Theorem 3.
LetE 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 thatf is not continuous at a∈E. Since F is locally compact, there is a compact neighbourhood V of fa such that fU⊄V for all neighbourhood U of a. By the local connectedness of E, there is a generalized sequence Uαα∈Λ of connected neighbourhoods of a, which is a basis of neighbourhoods of a in E. Then,(13)fa∈fUα∩V,fUα∩F∖V≠∅,for allα∈Λ.
Sincef satisfies the IVP, fUα is connected for all α∈Λ. By Theorem 1, there exists a generalized sequence xαα∈Λ such that(14)xα∈Uα,fxα∈FrV,for allα∈Λ.
By the compactness ofV, FrV is compact. Then, there exists a subsequence xαα∈Δ⊂Λ such that fxαα∈Δ converges to b∈FrV. Then, the generalized sequence xα,fxαα∈Δ converges to a,b. Since Γf is closed, then a,b∈Γf. Therefore, b=fa and fa∈FrV, which contradicts that V is a neighbourhood of fa. Thus, 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.Theorem 4.
Letf be a real valued function on an interval I of ℝ satisfying the IVP. If for all y∈ℝ,f−1y is closed in I, then f is continuous.Proposition 2.
Letf:E⟶F be a function; then:(1)
If the graph off is closed, then for every y∈F,f−1y is closed in E.(2)
If the graph off is sequentially closed, then for every y∈F,f−1y is sequentially closed in E.(3)
Iff is continuous, then for every y∈F,f−1y is closed in E.Proof.
(1)
Letxαα∈Λ be a generalized sequence in f−1y that converges to x. xα,yα∈Λ is a sequence in Γf that converges to x,y. Since Γf is closed, then fx=y. Hence, x∈f−1y. Therefore, f−1y is closed in E.(2)
In the same way, we show (2).(3)
The continuity off implies that the graph of f is closed, and then we have (3).Remark 3.
(1)
The reverse of 1 and 2 is false, in general. As in Example 2, sincef is injective, f−1y is closed in ℝ, but f has no closed graph.(2)
Theorem2 is a corollary of Theorem 4 and Proposition 2.The following theorem is a generalization of the real case in Theorem4.Theorem 5.
Suppose thatF 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 thatf is not continuous at a∈E. As in the proof of Theorem 2, there is r>0 and a sequence ann∈ℕ in E that converges to a, and for all n∈ℕ,fan∈FrV=S, where V=Bfa,randSfa,rare the ball and the sphere of radius fa, respectively. Hence, ann∈ℕ is in f−1S which is closed. Then, a∈f−1S and fa∈S: a contradiction. Thus, f is continuous.Theorem 6.
Suppose thatF 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.Proof.
LetS=Sa,r=x∈F:x−a=r be the sphere of center a and of radius r>0. Let xnn∈ℕ be a sequence in f−1S that converges to x. Let us show that x∈f−1S. We know that the sequence fxnn∈ℕ is in S. Since F is a finite dimensional normed space, then S is compact. Hence, there is a subsequence fxhnn∈ℕ of fxnn∈ℕ that converges to y∈S. Since the graph Γf of f is sequentially closed and xhn,fxhnn∈ℕ is a sequence in Γf that converges to x,y, then x,y in Γf and y=fx. Hence, fx∈S and x∈f−1S. Therefore, f−1S is sequentially closed in E.Theorem 7.
Suppose thatF 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.Proof.
LetS=Sa,r=x∈F:x−a=r be the sphere of center a and of radius r>0. Let x be in the closure f−1S¯; then, there exists a generalized sequence xαα∈Λ in f−1S that converges to x. Then, the sequence fxαα∈Λ is in S which is compact. Hence, there is a subsequence fxαα∈Δ⊂Λ that converges to y∈S. The graph Γf of f is closed and xα,fxαα∈Δ is in Γf; then, y=fx. Hence, fx∈S and x∈f−1S. Therefore, f−1S is closed in E.Corollary 2.
Suppose thatF 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.



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