Lipschitz function and differentiabily

Asked Nov 03 '13 at 20:46

Active Nov 03 '13 at 20:46

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I'm trying to prove this:

$U \subset \Bbb R^n$. Let $f:U\rightarrow\Bbb R$ be a differentiable function. Then

$\exists \;k\in\Bbb R, k\neq 0$ so that $\mid df_a(v)\mid \leq k \parallel v\parallel \;\forall \;a\in U, \forall \; v \in \Bbb R^n$ $\iff$ $f$ is Lipschitz

I haven't really tried much since I don't know where to start.

asked Nov 03 '13 at 20:46

Cath

1

I think you might want $k \in R$, not $k \in R^n$. – Robert Lewis Nov 03 '13 at 20:47
To get an idea of how the proof should go, first try proving it for $n=1$. For one direction, use the definition of derivative. For the other direction, use the mean value theorem. – Stephen Montgomery-Smith Nov 03 '13 at 21:26
OK, I've proven it for $n=1$ and also this side $\implies$ of the implication, but I can't do the other one... – Cath Nov 03 '13 at 23:32
I got the other side too. Thanks! – Cath Nov 03 '13 at 23:50

Lipschitz function and differentiabily

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