Definition:Piecewise Continuous Function/Bounded
< Definition:Piecewise Continuous Function(Redirected from Definition:Bounded Piecewise Continuous Function)
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Definition
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 $\closedint a b$.
Also see
- Piecewise Continuous Function with One-Sided Limits is Bounded
- Bounded Piecewise Continuous Function may not have One-Sided Limits