Lebesgue criterion for Riemann integrability

Lebesgue criterion for Riemann integrability: Let $f:[a,b]\to\mathbb{R}$ be a bounded function on the closed interval $[a,b]$ (see interval). Let $D_f\subseteq [a,b]$ be the set of points where $f$ is discontinuous. Then $f$ is Riemann integrable on $[a,b]$ if and only if $D_f$ is a null set with respect to Lebesgue measure on $\mathbb{R}$.

Equivalently, a bounded function is Riemann integrable exactly when it is continuous almost everywhere on $[a,b]$ with respect to Lebesgue measure.