Definition:Differentiable Mapping/Vector-Valued Function/Point/Definition 2

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Definition

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

Let $f : \mathbb X \to \R^m$ be a vector valued function.


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

$\ds \lim_{h \mathop \to \bszero} \frac {\norm {\map f {x + h} - \map f x - \map T h}} {\norm h} = 0$


Sources