Definition:Piecewise Continuous Function/Improper Integrals

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Let $f$ be a real function defined on a closed interval $\left[{a \,.\,.\, b}\right]$.

$f$ is piecewise continuous with improper integrals if and only if:

there exists a finite subdivision $\left\{{x_0, x_1, \ldots, x_n}\right\}$ of $\left[{a \,.\,.\, b}\right]$, where $x_0 = a$ and $x_n = b$, such that for all $i \in \left\{ {1, 2, \ldots, n}\right\}$:
$(1): \quad f$ is continuous on $\left({x_{i − 1} \,.\,.\, x_i}\right)$
$(2): \quad$ the improper integrals $\displaystyle \int_{ {x_{i - 1} }^+}^{ {x_i}^-} f \left({x}\right) \rd x$ all exist.

Also see