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The problem is:

Let g be of class $C^1$ on $\Delta$⊂$ℝ^n$ and K be a compact subset of Δ. Show that there is a number C such that |g(s)-g(t)|≤C|s-t| for every s,t∈K.


I have proved that it is true when K is convex, but I do not know how to proceed. Anyone could help me to solve this question?

Thanks

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  • $\begingroup$ is $\Delta$ open? or how are $C^1$ functions on $\Delta$ defined? $\endgroup$ – user251257 Sep 2 '15 at 11:54
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I suppose that $\Delta$ is an open subset of $\mathbb{R}^n$.

If $g$ is $C^1$ then it is locally lipschitz (to prove this fact you can use the strategy you have used in the case $K$ convex, that is the intermediate value theorem I think).

Now locally lipschitz functions on compact sets are globally lipschitz, and this can be proved by covering argument ( See if locally Lipschitz implies Lipschitz on compacts. )

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