Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Lipschitz function

  1. Apr 16, 2010 #1

    ibc

    User Avatar

    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
     
  2. jcsd
  3. Apr 16, 2010 #2
    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.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook