Set of discontinuities

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

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

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 $D(f)=[a,b]$.