Definition:Piecewise Continuously Differentiable Function/Definition 1

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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.


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