Continuity Points of Typical Bounded Functions

TL;DR

This paper explores the typical discontinuity behavior of bounded functions within a linear space, extending a known measure-theoretic result to a topological context and providing illustrative examples.

Contribution

It introduces a topological analogue of a measure-theoretic theorem on discontinuities in bounded functions and offers relevant examples.

Findings

Typical elements are discontinuous almost everywhere in the topological setting.

The paper establishes a topological version of Kostyrko and Salat's theorem.

Examples demonstrate the applicability of the topological analogue.

Abstract

Kostyrko and Salat showed that if a linear space of bounded functions has an element that is discontinuous almost everywhere, then a typical element in the space is discontinuous almost everywhere. We give a topological analogue of this theorem and provide some examples.