A Kuratowski theorem revisited

TL;DR

This paper revisits Kuratowski's problem about the continuity of functions with the Baire property, extending previous results to category bases and removing the separability condition.

Contribution

It generalizes Kunugi's result by addressing Kuratowski's question within the framework of category bases, removing the need for separability.

Findings

Extended the theorem to category bases

Removed the separability condition from Kuratowski's problem

Provided new insights into functions with the Baire property

Abstract

If $f : X\mapsto Y$ is a function having Baire property from a metric space $X$ into a separable metric space $Y$ , then $f$ is continuous except on a set of first category. Kuratowski asked whether the condition of separability could be removed. Several attempts were done in the past to solve this problem. In fact, the first impressive attempt was initiated by Kunugi. This paper is aimed towards solving the problem of Kuratowski in category bases which generalizes the result of Kunugi.