Definition:Piecewise Continuous Function/Improper Integrals

From ProofWiki
Jump to navigation Jump to search

Definition

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


$f$ is piecewise continuous with improper integrals 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 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


Sources