\begin{definition}[Definition:Differentiable Mapping/Vector-Valued Function/Region]
Let $\mathbb X$ be an open subset of $\R^n$. 
Let $f = \tuple {f_1, f_2, \ldots, f_m}^\intercal: \mathbb X \to \R^m$ be a vector valued function.
Let $S \subseteq \mathbb X$.
Then $f$ is '''differentiable in the open set $S$''' {{iff}} $f$ is differentiable at each $x$ in $S$.
This can be denoted $f \in \map {\CC^1} {S, \R^m}$.
\end{definition}