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.

Proof sketch: ($\Rightarrow$) If $f$ is integrable, choose a partition $P$ with $U(f,P)-L(f,P)$ small. Points of discontinuity must lie in subintervals where the oscillation of $f$ is not small; these subintervals can be shown to have arbitrarily small total length, yielding a measure-zero cover of $D$. ($\Leftarrow$) If $D$ has measure zero, cover $D$ by intervals of very small total length. On the remaining compact set , $f$ is continuous and hence uniformly continuous , so choose a fine partition there. Boundedness controls the contribution from the small cover of $D$, giving $U-L$ arbitrarily small.