Lebesgue criterion for Riemann integrability

Lebesgue criterion for Riemann integrability: Let $f:[a,b]\to\mathbb{R}$ be bounded , and let $D\subseteq[a,b]$ be the set of points where $f$ is discontinuous. Then $f$ is Riemann integrable on $[a,b]$ if and only if $D$ has measure zero ; i.e., for every $\varepsilon>0$ there exists a countable collection of open intervals $\{I_j\}$ such that $D\subseteq \bigcup_{j=1}^\infty I_j \quad\text{and}\quad \sum_{j=1}^\infty |I_j|<\varepsilon.$

This theorem is the complete structural characterization of Riemann integrability and explains exactly which discontinuity sets are allowed.