Definition:Piecewise Continuous Function

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

Then $f$ is piecewise continuous if it is bounded and there exists a finite partition $P = \left\{{x_0, \ldots, x_n}\right\}$ of $\left[{a \,.\,.\, b}\right]$ such that $f$ is continuous over $\left({x_{i−1}, \ldots, x_i}\right)$ for every $i \in \left\{{1, \ldots, n}\right\}$.