\begin{definition}[Definition:Continuously Differentiable/Real-Valued Function/Open Set]
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 $U$''' {{iff}}:
:$(1): \quad f$ is differentiable in $U$.
:$(2): \quad$ the partial derivatives of $f$ are continuous in $U$.
This can be denoted:
:$f \in \map {\CC^1} {\mathbb X, \R}$
\end{definition}