Returning functions with closed graph are continuous

Abstract

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.