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 $U$ 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:

$f \in \map {\CC^1} {\mathbb X, \R}$

Also see

• Definition:Differentiability Class for insight into the notation $\map {\CC^1} {U, \R}$