Mean value inequality

Builds on Differentiable function of several variables Euclidean space

Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}^m$ be differentiable on $U$ , and let $x, y \in U$ be points such that the segment $[x, y] = \{x + t (y - x) : 0 \le t \le 1\}$ lies in $U$ . Suppose that $M \ge 0$ and $\|Df(z)\| \le M$ for each $z \in [x, y]$ , where $\|Df(z)\| = \sup_{\|h\| \le 1} \|Df(z) h\|$ is the operator norm. Then $\|f(y) - f(x)\| \le M \, \|y - x\|.$

Remarks

For $m = 1$ , the inequality follows from the mean value theorem in several variables. For $m \ge 2$ there is in general no equality form, but the inequality still holds. A standard proof applies the one-variable mean value theorem to $\varphi(t) = (f(y) - f(x)) \cdot f(x + t(y - x))$ . On a convex set $C \subseteq U$ with $\|Df\| \le M$ on $C$ , the inequality holds for all $x, y \in C$ , so $f$ is Lipschitz on C.

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Euclidean space object
Function concept
Function of several variables concept
Limit of a function of several variables concept
Differentiable function of several variables definition
Mean value inequality statement

Where this leads

3 pages build on this one.

Tag	`02CYB184RC`
URI	`m2://02CYB184RC`
Kind	statement, theorem
Writing	W0 stub
MSC	26B05
Status	This copy of the site has no build data. Run `m2 status 02CYB184RC`.
Uses	Differentiable function of several variables, Euclidean space

Mean value inequality

Statement

Remarks

Reading path

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