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Lipschitz function

  1. Apr 16, 2010 #1


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    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?

  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:


    and refer to chapter 9.

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