Riemann integrability with finitely many discontinuities

Finite discontinuities imply Riemann integrability: Let $a<b$ and let $f:[a,b]\to\mathbb{R}$ be bounded. If the set of points where $f$ has a discontinuity is finite, then $f$ is a Riemann integrable function on $[a,b]$.

This gives a practical sufficient condition for Riemann integrability that can be checked by analyzing the set of discontinuities ; it is often proved using the oscillation criterion .