Definition:Differentiable Mapping/Real-Valued Function/Point/Definition 2
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The validity of the material on this page is questionable.
$\map f x + \map T h$ is a real number, so for the expression below to make sense, $\map r h\cdot h$ needs to be a real number, but $h$ is a vector and $\map r h$ is a real number, so $\map r h\cdot h$ is a vector (if $\cdot$ is supposed to denote scalar multiplication)
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues.
Definition
Let $U$ be an open subset of $\R^n$.
Let $f: U \to \R$ be a real-valued function.
Let $x \in U$.
$\map f x + \map T h$ is a real number, so for the expression below to make sense, $\map r h\cdot h$ needs to be a real number, but $h$ is a vector and $\map r h$ is a real number, so $\map r h\cdot h$ is a vector (if $\cdot$ is supposed to denote scalar multiplication)
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues.
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$f$ is differentiable at $x$ if and only if there exists a linear transformation $T: \R^n \to \R$ and a real-valued function $r: U \setminus \set x \to \R$ such that:
- $(1):\quad \map f {x + h} = \map f x + \map T h + \map r h \cdot h$
- $(2):\quad \displaystyle \lim_{h \mathop \to 0} \map r h = 0$