Proving Real Lipschitz Function Differentiability

[ibc]

Hello

I've been told that a (real) Lipschitz function (|f(x)-f(y)|<M|x-y|, for all x and y) must be differentiable almost everywhere.
but I don't see how I can prove it.
anyone has an idea?

Thanks

 

[snipez90]

A real lipschitz function is absolutely continuous and hence of bounded variation. These two statements are not too hard to prove. A real function of bounded variation has a finite derivative almost everywhere. This last statement is nontrivial and is a direct consequence of general facts about functions of bounded variation and more importantly the fact that a monotonic function over say a closed interval [a,b] has finite derivative almost everywhere (due to Lebesgue).

For proofs, see if the following link to Introductory Real Analysis by Komogorov and Fomin and Google Books is available:

http://books.google.com/books?id=z8...&resnum=2&ved=0CA8Q6AEwAQ#v=onepage&q&f=false

and refer to chapter 9.

Any text that talks about differentiation and the Lebesgue integral should have the required theorems.