Oscillation criterion for Riemann integrability

Oscillation criterion: Let $a<b$ and let $f:[a,b]\to\mathbb{R}$ be bounded. For a partition $P=\{a=x_0<x_1<\cdots<x_n=b\}$, let $\omega_i$ be the oscillation of $f$ on the subinterval $[x_{i-1},x_i]$. Then $f$ is a Riemann integrable function on $[a,b]$ if and only if for every $\varepsilon>0$ there exists a partition $P$ such that

This criterion is equivalent to the usual definition via upper sums and lower sums , and it is especially useful for proving integrability results like finite discontinuities imply integrability .