Definition:Continuously Differentiable
Definition
A differentiable function $f$ is continuously differentiable if and only if $f$ is of differentiability class $C^1$.
That is, if the first order derivative of $f$ (and possibly higher) is continuous.
Real Function
Let $I\subset\R$ be an open interval.
Then $f$ is continuously differentiable on $I$ if and only if $f$ is differentiable on $I$ and its derivative is continuous on $I$.
Real-Valued Function
Let $U$ be an open subset of $\R^n$.
Let $f: U \to \R$ be a real-valued function.
Then $f$ is continuously differentiable in the open set $U$ if and only if:
- $(1): \quad f$ is differentiable in $U$.
- $(2): \quad$ the partial derivatives of $f$ are continuous in $U$.
Vector-Valued Function
Let $U \subset \R^n$ be an open set.
Let $f: U \to \R^m$ be a vector-valued function.
Then $f$ is continuously differentiable in $U$ if and only if $f$ is differentiable in $U$ and its partial derivatives are continuous in $U$.
Also see
Sources
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): continuously differentiable