Definition:Piecewise Continuous Function/Variant 1

Definition
Let $\closedint a b$ be a closed interval.

Let $\set {x_0, x_1, \ldots, x_n}$ be a finite subdivision of $\closedint a b$, where $x_0 = a$ and $x_n = b$.

Let $f$ be a real function defined on $\closedint a b \setminus \set {x_0, x_1, \ldots, x_n}$.

$f$ is piecewise continuous with one-sided limits :


 * for all $i \in \set {1, 2, \ldots, n}$:


 * $(1): \quad f$ is continuous on $\openint {x_{i − 1} } {x_i}$


 * $(2): \quad$ the one-sided limits $\displaystyle \lim_{x \mathop \to {x_{i − 1} }^+} \map f x$ and $\displaystyle \lim_{x \mathop \to {x_i}^-} \map f x$ exist.