Definition:Piecewise Continuous Function/Bounded

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

$f$ is a bounded piecewise continuous function :


 * 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 $\closedint a b$.

Also see

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


 * Bounded Piecewise Continuous Function has Improper Integrals
 * Piecewise Continuous Function with Improper Integrals may not be Bounded


 * Bounded Piecewise Continuous Function is Darboux Integrable