Show function with infinitely many discontinuities is Riemann-integrable

As mentioned in Michael Greinecker's comment, a nice way to go is to use the following result:

Theorem: Let $f: [a,b] \rightarrow \mathbb{R}$ be a bounded function. If for all $c \in (a,b)$ the restriction of $f$ to $[c,b]$ is Riemann integrable, then $f$ is Riemann integrable on $[a,b]$ and $\int_a^b f = \lim_{c \rightarrow a^+} \int_c^b f$.

This result is proved in $\S 8.3.2$ of my honors calculus notes.