Definition:Piecewise Continuous Function/Improper Integrals

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

Then $f$ is piecewise continuous iff:

there exists a finite subdivision $\left\{{x_0, \ldots, x_n}\right\}$ of $\left[{a \,.\,.\, b}\right]$, $x_0 = a$ and $x_n = b$, such that:


 * $(1): \quad$ $f$ is continuous on $\left({x_{i−1} \,.\,.\, x_i}\right)$ for every $i \in \left\{{1, \ldots, n}\right\}$


 * $(2): \quad$ the improper integrals $\displaystyle \int_{x_{i-1}+}^{x_i-} f \left(x\right) \ \mathrm d x$ exist for every $i \in \left\{{1, \ldots, n}\right\}$.