Definition:Piecewise Continuous Function/One-Sided Limits

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

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


 * 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 one-sided limits $\displaystyle \lim_{x \mathop \to {x_{i − 1} }^+} f \left({x}\right)$ and $\displaystyle \lim_{x \mathop \to {x_i}^-} f \left({x}\right)$ exist.

Also known as
Some sources hyphenate: piecewise-continuous.

Some sources refer to condition $(2)$ as that $f \left({x}\right)$ is finite at the endpoints, but demands more rigor in its use of the term finite.

The one-sided limits can also be seen denoted as:
 * $f \left({x_{i − 1} + 0}\right)$ and $f \left({x_i - 0}\right)$

Also see

 * Piecewise Continuous Function with One-Sided Limits is Bounded
 * Bounded Piecewise Continuous Function may not have One-Sided Limits


 * Piecewise Continuous Function with One-Sided Limits is Riemann Integrable


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