Definition:Piecewise Continuously Differentiable Function/Closed Intervals

Definition

Let $f$ be a real function defined on a closed interval $\closedint a b$.

$f$ is piecewise continuously differentiable if and only if:

there exists a finite subdivision $\set {x_0, \ldots, x_n}$ of $\closedint a b$, $x_0 = a$ and $x_n = b$, such that:

$f$ is continuously differentiable on $\closedint {x_{i − 1} } {x_i}$, where the derivative at $x_{i − 1}$ understood as right-handed and the derivative at $x_i$ understood as left-handed, for every $i \in \set {1, \ldots, n}$.

Also see

• Piecewise Continuously Differentiable Function/Definition 2 is Continuous

Sources

• 1991: Reinhold Remmert: Theory of Complex Functions (2nd ed.): Part $\text{B}$, Ch. $6$: $\S 1.1$