Definition:Piecewise Continuously Differentiable Function

Definition
A real function $f$ defined on a closed interval $\left[{a \,.\,.\, b}\right]$ is piecewise continuously differentiable if:


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