Suppose that $g:[a,b]\to\mathbb{R}$ is continuous except at $r_1,r_2\in(a,b)$. Prove that $g$ is Riemann integrable on $[a,b]$.
Intuitively, I know this is true because the upper sum and lower sum will only differ by the contribution from $f(r_1)$ and $f(r_2)$, and you refine the partition until it disappears. But I'm not quite sure how to turn that into a formal proof.