Definition:Differentiable Mapping/Vector-Valued Function/Region

Definition
Let $\mathbb X$ be an open rectangle of $\R^n$.

Let $f = \left({f_1, f_2, \ldots, f_m}\right): \mathbb X \to \R^m$ be a vector valued function. Let $S \subseteq \mathbb X$.

Then $f$ is differentiable in the region $S$ iff $f$ is differentiable at each $x$ in $S$.

This can be denoted $f \in \mathcal C^1 \left({S, \R^m}\right)$.

Also see

 * Differentiability Class for insight into the notation $\mathcal C^1 \left({S, \R^m}\right)$.