# Definition:Continuously Differentiable/Real-Valued Function/Open Set

< Definition:Continuously Differentiable | Real-Valued Function(Redirected from Definition:Continuously Differentiable Real-Valued Function on Open Set)

## Definition

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 $U$.

This can be denoted:

- $f \in \mathcal C^1 \left({\mathbb X, \R}\right)$

## Also see

- Definition:Differentiability Class for insight into the notation $\mathcal C^1 \left({U, \R}\right)$