User:Ivar Sand/Sandbox

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

$f$ is piecewise continuous :

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 one-sided limits $\displaystyle \lim_{x \to x_{i−1^+}} f\left({x}\right)$ and $\displaystyle \lim_{x \to x_{i^-}} f\left({x}\right)$ exist for every $i \in \left\{{1, \ldots, n}\right\}$.

Theorems

 * Piecewise Continuous Function with One-Sided Limits is Bounded
 * Piecewise Continuous Function with One-Sided Limits is Riemann Integrable
 * Piecewise Continuous Function with One-Sided Limits is Uniformly Continuous on Each Piece