Continuity of Functions Relative to their Sets of Discontinuity

Abstract

It is well known that the set of points of continuity of a function $f\colon \mathbb R\to \mathbb R$ is a $G_\delta$ set, and that for every $F_\sigma$ set $D$ there is an $f$ such that the set of points of discontinuities of $f$ is $D$. In this paper we decompose $D$ into the union of two disjoint $F_\sigma$ sets $U$ and $V$, such that $f_{|V}$ is discontinuous on $V$, and we show that $V$ is maximal in the sense that there is an $f$ such that $f_{|\overline{D}}$ is continuous on $U$.