# Definition:Continuously Differentiable

## Contents

## 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 $U$.