# Definition:Piecewise Continuously Differentiable Function/Definition 1

## Definition

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

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

- $(1): \quad f$ 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 $\displaystyle \lim_{x \mathop \to {x_{i - 1}}^+} \map {f'} x$ and $\displaystyle \lim_{x \mathop \to {x_i}^-} \map {f'} x$ exist for every $i \in \set {1, \ldots, n}$.

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

## Sources

- 2008: David C. Ullrich:
*Complex Made Simple*: Part $\text{1}$, Ch. $2$