# Definition:Differentiable Mapping/Vector-Valued Function/Region

## Definition

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 $S$ if and only if $f$ is differentiable at each $x$ in $S$.

This can be denoted $f \in \map {\CC^1} {S, \R^m}$.