Definition:Differentiable Mapping/Vector-Valued Function

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Let $m,n\geq1$ be natural numbers.

At a Point

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

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

Definition 1

$f$ is differentiable at $x \in \R^n$ if and only if there exists a linear transformation $T:\R^n \to \R^m$ and a mapping $r : U \to \R^m$ such that:

$(1):\quad$ $\displaystyle f \left({x + h}\right) = f \left({x}\right) + T(h) + r\left({h}\right)\cdot \Vert h\Vert$
$(2):\quad$ $\displaystyle\lim_{h\to 0} r(h) = 0$.

Definition 2

$f$ is differentiable at $x \in \R^n$ if and only if for each real-valued function $f_j: j = 1, 2, \ldots, m$:

$f_j: \mathbb X \to \R$ is differentiable at $x$.

In an Open Set

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