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Homework Help: Help interpreting HW question on Lipschitz Hölder

  1. Sep 9, 2012 #1
    1. The problem statement, all variables and given/known data
    I only need help interpreting the following:
    Show that every Lipschitz continuous function is α-Hölder continuous for
    every α ∈ (0, 1
    The definition of both is given in the homework so this seems trivial but it's a graduate level class. Am I mising something? Thanks for any help!

    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Sep 9, 2012 #2

    HallsofIvy

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    Well, what are those definitions? Why do you say this is "trivial"?
     
  4. Sep 9, 2012 #3
    Well by the definitions, Lipschitz is a special case of α-Hölder when α=1. Since α is contained in the interval (0,1] (which is the interval given for α-Hölder) then by def. every Lipschitz continuous function is α-Hölder continuous.
     
  5. Sep 9, 2012 #4
    Your question asked to show that it is α-Hölder continuous for every α ∈ (0, 1], not just for α=1. Unless this was a typo? Yes, Lipschitz implies Hölder of order 1. But does it imply this for all orders less than 1?
     
  6. Sep 9, 2012 #5
    Ahhh, great! yes your right I see it now. Funny how sometimes one can not see what is right in front of them. Thanks for the help, that's exactly what I needed.
     
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