Set of discontinuities

A set of discontinuities of a function $f:E\to\mathbb R$ , where $E\subseteq\mathbb R$ , is the set

D(f)=\{x\in E : f \text{ is not continuous at } x\}.

Equivalently, $x\in D(f)$ exactly when $x$ is a discontinuity point of $f$ .

Remarks

This set is central in Riemann integrability; for bounded functions on an interval, integrability can be characterized in terms of how “small” $D(f)$ is (see the Lebesgue criterion for Riemann integrability).

Examples

If $f$ is continuous on $[a,b]$ , then $D(f)$ is the empty set.
On $[a,b]$ , let $f$ be the indicator function of $\mathbb Q\cap[a,b]$ . Then $D(f)=[a,b]$ .