Definition:Piecewise Continuously Differentiable Function
Definition
Let $f$ be a real function defined on a closed interval $\closedint a b$.
Definition 1
$f$ 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 $\ds \lim_{x \mathop \to {x_{i - 1}}^+} \map {f'} x$ and $\ds \lim_{x \mathop \to {x_i}^-} \map {f'} x$ exist for every $i \in \set {1, \ldots, n}$.
Definition 2
$f$ 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
Also defined as
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Other definitions of Piecewise Continuously Differentiable Function exist. Some examples are shown in the list below. The list is condensed; see the discussion page for a detailed description.
- $(2)$ is replaced by: $f$ is continuously differentiable on the intervals $\openint {x_{i − 1} } {x_i}$.
- $(1)$ is replaced by: $f$ is piecewise continuous, and $(2)$ is replaced by:
- $f$ is continuously differentiable on the intervals $\openint {x_{i − 1} } {x_i}$ and $f'$ has one-sided limit(s) at every $x_i$, or
- $f$ is continuously differentiable on the intervals $\openint {x_{i − 1} } {x_i}$ and $f'$ has one-sided limit(s) at every $x_i$, and, in addition, $f$ is allowed to be undefined at the points $x_i$, or
- $f$ is continuously differentiable on the intervals $\openint {x_{i − 1} } {x_i}$, and $f'$ is bounded on $\openint {x_{i − 1} } {x_i}$.
- The codomain of $f$ is $\C$ instead of $\R$.
Also known as
A piecewise continuously differentiable function is referred to in some sources as a piecewise smooth function.
However, as a smooth function is defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ as being of differentiability class $\infty$, this can cause confusion, so is not recommended.