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Returning functions with closed graph are continuous

  • Taras Banakh EMAIL logo , Małgorzata Filipczak and Julia Wódka
Published/Copyright: March 10, 2020
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 70 Issue 2

Abstract

A function f : X → ℝ defined on a topological space X is called returning if for any point x ∈ X there exists a positive real number Mx such that for every path-connected subset Cx ⊂ X containing the point x and any y ∈ Cx ∖ {x} there exists a point z ∈ Cx ∖ {x, y} such that |f(z)| ≤ max{Mx, |f(y)|}. A topological space X is called path-inductive if a subset U ⊂ X 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.

Keywords: continuous function; returning function; closed graph; path-inductive space
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

© 2020 Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

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  4. Wild sets in global function fields
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  7. Returning functions with closed graph are continuous
  8. On sets of points of approximate continuity and ϱ-upper continuity
  9. Investigation of the fifth Hankel determinant for a family of functions with bounded turnings
  10. On solvability of some nonlocal boundary value problems for biharmonic equation
  11. The study of piecewise pseudo almost periodic solutions for impulsive Lasota-Wazewska model with discontinuous coefficients
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  16. Modified Lupaş-Jain operators
  17. New fixed point results in bv(s)-metric spaces
  18. Improved Young and Heinz operator inequalities for unitarily invariant norms
  19. Scrutiny of some fixed point results by S-operators without triangular inequality
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https://doi.org/10.1515/ms-2017-0352
Keywords for this article
continuous function; returning function; closed graph; path-inductive space

Articles in the same Issue

  1. Regular papers
  2. Relatively residuated lattices and posets
  3. Strongly s-dense injective hull and Banaschewski’s theorems for acts
  4. Wild sets in global function fields
  5. Generators and integral points on elliptic curves associated with simplest quartic fields
  6. New Filbert and Lilbert matrices with asymmetric entries
  7. Returning functions with closed graph are continuous
  8. On sets of points of approximate continuity and ϱ-upper continuity
  9. Investigation of the fifth Hankel determinant for a family of functions with bounded turnings
  10. On solvability of some nonlocal boundary value problems for biharmonic equation
  11. The study of piecewise pseudo almost periodic solutions for impulsive Lasota-Wazewska model with discontinuous coefficients
  12. Strongly increasing solutions of higher-order quasilinear ordinary differential equations
  13. Oscillation of second order half-linear neutral differential equations with weaker restrictions on shifted arguments
  14. Filippov solutions of vector Dirichlet problems
  15. Sequences of positive homoclinic solutions to difference equations with variable exponent
  16. Modified Lupaş-Jain operators
  17. New fixed point results in bv(s)-metric spaces
  18. Improved Young and Heinz operator inequalities for unitarily invariant norms
  19. Scrutiny of some fixed point results by S-operators without triangular inequality
  20. The lattices of families of regular sets in topological spaces
  21. Iterated partial summations applied to finite-support discrete distributions
  22. Hamiltonicity of a class of toroidal graphs
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