\begin{definition}[Definition:Piecewise Continuous Function/Variant 1]
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''' {{iff}}:
: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 $\ds \lim_{x \mathop \to {x_{i − 1} }^+} \map f x$ and $\ds \lim_{x \mathop \to {x_i}^-} \map f x$ exist.
\end{definition}