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

< Definition:Differentiable Mapping | Vector-Valued Function(Redirected from Definition:Differentiable Vector-Valued Function at Point)

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## 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.

### 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:

- $(1): \quad \map f {x + h} = \map f x + \map T h + \map r h \cdot \norm h$

- $(2): \quad \ds \lim_{h \mathop \to 0} \map r h = 0$

### Definition 2

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

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

### Definition 3

$f$ 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 $x$.

## Also see

## Sources

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- 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards:
*Calculus*(8th ed.): $\S 2.1$, $13.4$