Finitely many discontinuities implies Riemann integrable

Finitely many discontinuities implies Riemann integrable: Let $f:[a,b]\to\mathbb{R}$ be bounded . If $f$ has only finitely many discontinuities on $[a,b]$, then $f$ is Riemann integrable on $[a,b]$.

This theorem illustrates that Riemann integration tolerates “isolated bad points.” It is a stepping stone toward the full characterization via the size of the discontinuity set.