Let $a,b \in \mathbb{R}$ with $a < b$, and let $f: [a,b] \to \mathbb{R}$ be piecewise continuous in the following sense: there exists a finite partition
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\begin{align*}
a = x_0 < x_1 < \cdots < x_n = b
\end{align*}
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such that, for each $k \in \{1,\dots,n\}$, the restriction $f|_{(x_{k-1},x_k)}$ extends to a [continuous function](/page/Continuous%20Function)