Piecewise Continuous Function with One-Sided Limits is Darboux Integrable/Proof 1

Proof
We are given that $f$ is piecewise continuous with one-sided limits on $\left[{a \,.\,.\, b}\right]$.

From Piecewise Continuous Function with One-Sided Limits is Bounded, $f$ is a bounded piecewise continuous function.

The result follows from Bounded Piecewise Continuous Function is Riemann Integrable.