User:Ivar Sand/Sandbox

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

$f$ is piecewise continuous :

there exists a finite subdivision $\set {x_0, \ldots, x_n}$ of $\closedint a b$, $x_0 = a$ and $x_n = b$, such that:
 * $(1): \quad$ $f$ is continuous on $\openint {x_{i − 1} } {x_i}$ for every $i \in \set {1, \ldots, n}$
 * $(2): \quad$ the one-sided limits $\ds \lim_{x \mathop \to x_{i − 1^+} } \map f x$ and $\ds \lim_{x \mathop \to x_{i^-} } \map f x$ exist for every $i \in \set {1, \ldots, n}$.

Theorems

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