On piecewise continuous mappings of metrizable spaces

TL;DR

This paper proves that resolvable-measurable functions from metrizable spaces to regular spaces are piecewise continuous, and characterizes resolvable-measurability as equivalent to piecewise continuity in metrizable completely Baire spaces.

Contribution

It establishes a new equivalence between resolvable-measurability and piecewise continuity for functions on specific metrizable spaces.

Findings

Resolvable-measurable functions are piecewise continuous from metrizable to regular spaces.

In metrizable completely Baire spaces, resolvable-measurability is equivalent to piecewise continuity.

The results extend understanding of function regularity in metrizable spaces.

Abstract

Let $f \colon X \rightarrow Y$ be a resolvable-measurable mapping of a metrizable space $X$ to a regular space $Y$. Then $f$ is piecewise continuous. Additionally, for a metrizable completely Baire space $X$, it is proved that $f$ is resolvable-measurable if and only if it is piecewise continuous.