\begin{definition}[Definition:Piecewise Continuously Differentiable Function/Definition 1]
Let $f$ be a real function defined on a closed interval $\closedint a b$.
$f$ is '''piecewise continuously differentiable''' {{iff}}:
:$(1): \quad f$ is continuous
:$(2): \quad$ there exists a finite subdivision $\set {x_0, \ldots, x_n}$ of $\closedint a b$, $x_0 = a$ and $x_n = b$, such that:
::$(2.1): \quad f$ is continuously differentiable on $\openint {x_{i - 1}} {x_i}$ for every $i \in \set {1, \ldots, n}$
::$(2.2): \quad$ 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 $i \in \set {1, \ldots, n}$.
Note that $f'$ is piecewise continuous with one-sided limits.
\end{definition}