Rademacher's Theorem: Introduction to Lipschitz Continuity

[jamilmalik]

Hello everyone,

I was wondering if I could get a simple introduction to this Theorem since I will have to be giving a presentation on it within the next month. Based on the statement itself, there is an assumption made in the hypothesis which is something I haven't quite understood yet:

If $U$ is an open subset of $\mathbb{R^n}$ and $f:U \to \mathbb{R^m}$ is Lipschitz continuous, then $f$ is differentiable almost everywhere in $U$; that is, the points in $U$ at which $f$ is not differentiable form a set of Lebesgue measure zero.

What exactly is Lipschitz continuous? I asked a professor of mine and he said to think about it as a bounded slope which makes sense looking at the definition of Lipschitz continuity. However, could someone please provide a thorough explanation of this? For instance, how would one go about proving this Theorem?

 

[Greg Bernhardt]

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?