Definition:Piecewise Continuous Function/Improper Integrals

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

$f$ is piecewise continuous with improper integrals :


 * 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 for all $i \in \set {1, 2, \ldots, n}$:


 * $(1): \quad f$ is continuous on $\openint {x_{i − 1} } {x_i}$


 * $(2): \quad$ the improper integrals $\ds \int_{ {x_{i - 1} }^+}^{ {x_i}^-} \map f x \rd x$ all exist.

Also see

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


 * Piecewise Continuous Function does not necessarily have Improper Integrals