Definition:Piecewise Continuously Differentiable Function

Definition
Let $f$ be a real function defined on a closed interval $\left[{a \,.\,.\, b}\right]$.

Then $f$ is piecewise continuously differentiable iff:


 * $(1): \quad$ $f$ is continuous
 * $(2): \quad$ there exists a finite subdivision $\left\{{x_0, \ldots, x_n}\right\}$ of $\left[{a \,.\,.\, b}\right]$, $x_0 = a$ and $x_n = b$, such that $f$ is continuously differentiable on $\left({x_{i−1} \,.\,.\, x_i}\right)$ and the one-sided limits $\displaystyle \lim_{x \to x_{i−1^+}} f′\left({x}\right)$ and $\displaystyle \lim_{x \to x_{i^-}} f′\left({x}\right)$ exist for every $i \in \left\{{1, \ldots, n}\right\}$.

Note that $f′$ is piecewise continuous.

Extendability Theorem for Derivatives Continuous on Open Intervals‎ gives that the following definition is equivalent to the definition above and is therefore an alternative formulation of that definition:


 * $(1'): \quad$ $f$ is continuous
 * $(2'): \quad$ there exists a finite subdivision $\left\{{x_0, \ldots, x_n}\right\}$ of $\left[{a \,.\,.\, b}\right]$, $x_0 = a$ and $x_n = b$, such that $f$ is continuously differentiable on $\left[{x_{i−1} \,.\,.\, x_i}\right]$, the derivatives at $x_{i−1}$ and $x_i$ understood as one-sided derivatives, for every $i \in \left\{{1, \ldots, n}\right\}$.

Note that $(1′)$ can be removed from this definition since the continuity of $f$ follows from $(2′)$ because by:
 * Right-Hand Differentiable Function is Right-Continuous, $f$ is continuous at $a$
 * Left-Hand Differentiable Function is Left-Continuous, $f$ is continuous at $b$
 * Left-Hand and Right-Hand Differentiable Function is Continuous, $f$ is continuous at every $x_i$, $i \in \left\{{1, \ldots, n-1}\right\}$
 * Differentiable Function is Continuous, $f$ is continuous on every interval $\left({x_{i−1} \,.\,.\, x_i}\right)$, $i \in \left\{{1, \ldots, n}\right\}$.

Also defined as
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 ($x_{i−1} \,.\,.\, x_i$).

- $(1)$ is replaced by: $f$ is piecewise continuous, and $(2)$ is replaced by: - The codomain of $f$ is $\C$ instead of $\R$.
 * $f$ is continuously differentiable on the intervals ($x_{i−1} \,.\,.\, x_i$) and $f′$ has one-sided limit(s) at every $x_i$, or
 * $f$ is continuously differentiable on the intervals ($x_{i−1} \,.\,.\, x_i$) 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 ($x_{i−1} \,.\,.\, x_i$), and $f′$ is bounded on ($x_{i−1} \,.\,.\, x_i$).