Definition:Piecewise Continuously Differentiable Function

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

Also see

 * Equivalence of Definitions of Piecewise Continuously Differentiable Function


 * Definition:Piecewise Continuous Function

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 $\left( {x_{i−1} \,.\,.\, x_i} \right)$.

- $(1)$ 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 $\left( {x_{i−1} \,.\,.\, x_i} \right)$ and $f'$ has one-sided limit(s) at every $x_i$, or
 * $f$ is continuously differentiable on the intervals $\left( {x_{i−1} \,.\,.\, x_i} \right)$ 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 $\left( {x_{i−1} \,.\,.\, x_i} \right)$, and $f'$ is bounded on $\left( {x_{i−1} \,.\,.\, x_i} \right)$.

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 as being of differentiability class $\infty$, this can cause confusion, so is not recommended.