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

Theorem

Let $f$ be a real function defined on a closed interval $\closedint a b$.

Let $f$ be piecewise continuous with one-sided limits on $\closedint a b$.

Then $f$ is Darboux integrable on $\closedint a b$.

Proof

We are given that $f$ is piecewise continuous with one-sided limits on $\closedint a b$.

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 Darboux Integrable.

$\blacksquare$