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Question about Lipschitz continuity

  1. Jul 18, 2009 #1

    quasar987

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    1. The problem statement, all variables and given/known data
    This should be easy but I'm stomped.

    Let K be a compact set in a normed linear space X and let f:X-->X be locally Lipschitz continuous on X. Show that there is an open set U containing K on which f is Lipschitz continuous.


    2. Relevant equations
    locally Lipschitz means that for every x in X, there is a nbdh around x on which f is Lipschitz.


    3. The attempt at a solution
    The obvious thing to do it seems it take an open cover of K by sets on which f is Lipschitz continuous and extract a finite subcover. But then what?!?
     
  2. jcsd
  3. Jul 18, 2009 #2

    Dick

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    Wouldn't it be nice if you could make sure that the final open set containing K and covered by the subcover, were convex? Replace K by it's convex hull, it's still compact, right? Then take the finite subcover. Then pick a open set just a 'little' bigger than K, but still convex and contained in the subcover. That works, doesn't it?
     
    Last edited: Jul 18, 2009
  4. Jul 18, 2009 #3

    quasar987

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    Of course, yes! Thanks Dick.
     
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