Strict differentiability

Builds on Fréchet derivative Differentiable function of several variables

Definition

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}^m$ , and let $a \in U$ . The function $f$ is strictly differentiable at $a$ if there is a linear map $L \colon \mathbb{R}^n \to \mathbb{R}^m$ such that $\lim_{\substack{(x, y) \to (a, a) \\ x \ne y}} \frac{\|f(x) - f(y) - L(x - y)\|}{\|x - y\|} = 0.$

Remarks

If $f$ is strictly differentiable at $a$ , put $y = a$ : then $f$ is differentiable at $a$ and $L = Df(a)$ is its Fréchet derivative. The converse is false. The function $f(x) = x^2 \sin(1/x)$ for $x \ne 0$ , $f(0) = 0$ , is differentiable at $0$ , but it is not strictly differentiable at $0$ because $f'$ is not continuous at $0$ . A function of class $C^1$ on $U$ is strictly differentiable at each point of $U$ . The same definition works for maps between Banach spaces, with norms in place of Euclidean norms.

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Euclidean space object
Function concept
Fréchet derivative definition
Function of several variables concept
Limit of a function of several variables concept
Differentiable function of several variables definition
Strict differentiability definition

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C^1 functions are strictly differentiable

Tag	`0146MBQNNE`
URI	`m2://0146MBQNNE`
Kind	definition
Writing	W0 stub
MSC	26B05
Defines	Strict differentiability
Uses	Fréchet derivative, Differentiable function of several variables

Strict differentiability

Definition

Remarks

Reading path

Where this leads

Further reading

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