Abstract :

Suppose $D(f)$ denotes the set of points of discontinuity of a function $f$ from a topological space $X$ to the real line. The questions we want to address are the following: 1. When $X=\mathbb{R}$ (the real line), can $D(f)$ be the set of irrationals? 2. In general, can we describe $D(f)$?