A simple question on the Lipschitz property

Asked Aug 25 '15 at 20:06

Active Aug 25 '15 at 20:18

Viewed 47 times

Suppose $f:\mathbb{R}^n\to\mathbb{R}$ is differentiable and $L$-Lipschitz, namely

$$|f(x) - f(y)|\leq L ||x-y||_2 \ \ \forall x,y\in\mathbb{R}^n~.$$

How does this imply

$$||\nabla f||_2\leq L\ ?$$

It's clear when $n=1,$ but how can it be shown for higher $n?$

asked Aug 25 '15 at 20:06

Hedonist

1

You can still use the mean value theorem. – Augustin Aug 25 '15 at 20:10
4

Look at $f(x + t\nabla f(x)) - f(x)$ for small $\lvert t\rvert$. – Daniel Fischer Aug 25 '15 at 20:13
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What makes you think it's a silly question? – Daniel Fischer Aug 25 '15 at 20:15
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@Hedonist there's no such thing. – null Aug 25 '15 at 20:15
Only meant to say I should have thought a bit more before posting. – Hedonist Aug 25 '15 at 20:15

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