Definition:Continuously Differentiable/Real-Valued Function/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 $U$ :
 * $(1): \quad f$ 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 $\mathcal C^1 \left({U, \R}\right)$