# Definition:Piecewise Continuous Function/Bounded

## Definition

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

$f$ is a **bounded piecewise continuous function** if and only if:

- there exists a finite subdivision $\left\{{x_0, x_1, \ldots, x_n}\right\}$ of $\left[{a \,.\,.\, b}\right]$, where $x_0 = a$ and $x_n = b$, such that:

- $(1): \quad$ for all $i \in \left\{{1, 2, \ldots, n}\right\}$, $f$ is continuous on $\left({x_{i − 1} \,.\,.\, x_i}\right)$

- $(2): \quad$ $f$ is bounded on $\left[{a \,.\,.\, b}\right]$.

## Also see

- Piecewise Continuous Function with One-Sided Limits is Bounded
- Bounded Piecewise Continuous Function may not have One-Sided Limits