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

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## Theorem

Let $f$ be a real function defined on a closed interval $\left[{a \,.\,.\, b}\right]$.

Let $f$ be piecewise continuous with one-sided limits on $\left[{a \,.\,.\, b}\right]$.

Then $f$ is Riemann integrable on $\left[{a \,.\,.\, b}\right]$.

## 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.

$\blacksquare$