Definition:Differentiable Mapping/Vector-Valued Function

At a Point
Let $f = (f_1,\ldots,f_m) : \mathbb X \to \R^m$ where $\mathbb X \subseteq \R^n$ be a vector valued function.

Then $f$ is differentiable at $x \in \R^n$ if for each $j = 1, \ldots, m$, $f_j: \mathbb X \to \R$ is differentiable as a real valued function.

In a Region
Let $S \subseteq \mathbb X$.

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

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