Definition:Piecewise Continuous Function/Bounded

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

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

there exists a finite subdivision $\set {x_0, x_1, \ldots, x_n}$ of $\closedint a b$, where $x_0 = a$ and $x_n = b$, such that:
$(1): \quad$ for all $i \in \set {1, 2, \ldots, n}$, $f$ is continuous on $\openint {x_{i − 1} } {x_i}$
$(2): \quad$ $f$ is bounded on $\closedint a b$.

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