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Returning functions with closed graph are continuous

Taras Banakh, Małgorzata Filipczak and Julia Wódka
From the journal Mathematica Slovaca

Abstract

A function f : X → ℝ defined on a topological space X is called returning if for any point xX there exists a positive real number Mx such that for every path-connected subset CxX containing the point x and any yCx ∖ {x} there exists a point zCx ∖ {x, y} such that |f(z)| ≤ max{Mx, |f(y)|}. A topological space X is called path-inductive if a subset UX is open if and only if for any path γ : [0, 1] → X the preimage γ–1(U) is open in [0, 1]. The class of path-inductive spaces includes all first-countable locally path-connected spaces and all sequential locally contractible spaces. We prove that a function f : X → ℝ defined on a path-inductive space X is continuous if and only if it is returning and has closed graph. This implies that a (weakly) Świątkowski function f : ℝ → ℝ is continuous if and only if it has closed graph, which answers a problem of Maliszewski, inscribed to Lviv Scottish Book.

MSC 2010: 26A15; 54C08; 54D05

  1. Communicated by Tomasz Natkaniec

Acknowledgement

The authors express their sincere thanks to the referees for fruitful comments and suggestions of improving the presentations and adding some new results and examples.

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Received: 2019-03-05
Accepted: 2019-08-27
Published Online: 2020-03-10
Published in Print: 2020-04-28

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