Definition:Piecewise Continuously Differentiable Function/Definition 1

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

$f$ is piecewise continuously differentiable :


 * $(1): \quad$ $f$ is continuous
 * $(2): \quad$ there exists a finite subdivision $\left\{{x_0, \ldots, x_n}\right\}$ of $\left[{a \,.\,.\, b}\right]$, $x_0 = a$ and $x_n = b$, such that:
 * $(2.1): \quad$ $f$ is continuously differentiable on $\left({x_{i − 1} \,.\,.\, x_i}\right)$ for every $i \in \left\{{1, \ldots, n}\right\}$
 * $(2.2): \quad$ 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 for every $i \in \left\{{1, \ldots, n}\right\}$.

Note that $f'$ is piecewise continuous with one-sided limits.