Is $f(x)$ Riemann integrable on $[0,2]?$ Yes/No

Theorem:- If $f:[a,b]\to\mathbb{R}$ be bounded on $[a.b]$ and is continuous on $[a,b]$ except for a finite number of points . Then $f$ is Riemann Integrable. (More generally if the derived set of the set of discontinuities is finite , then also the function is Riemann integrable. More generally if the set of discontinuities is a set of measure zero. Then the function is Riemann integrable.)

For a proof.Try and enclose the points of discontinuity by non overlapping subintervals whose lenght is lesser than a given epsilon and cover the rest intervals by partitions such that $U(P,f)-L(P,f)\leq \epsilon$. This is possible as in the rest of those intervals f is continuious and hence Riemann integrable. This will give you the required partition of the set $[a,b]$.