Definition:Piecewise Continuously Differentiable Function/One-Sided Limits

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, for every $i \in \set {1, \ldots, n}$:

$(1): \quad f$ is continuously differentiable on $\openint {x_{i - 1}} {x_i}$

$(2): \quad$ 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.

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

Sources

• 2008: David C. Ullrich: Complex Made Simple: Part $1$, Ch. $2$

(This definition has an additional continuity requirement.)