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Derivatives and continuity / Lipschitz equation

  1. Dec 3, 2012 #1
    Hi! I think I've managed to solve this problem, but I'd like it to be checked

    1. The problem statement, all variables and given/known data

    show that if $$f : A\subset \mathbb{R}\to \mathbb{R}$$ and has both right derivative:
    $$f_{+}'(x_0),$$

    and left derivative
    $$f_{-}'(x_0)$$
    in $$x_0\in A$$, then $$f$$
    is continuos in
    $$x_0.$$

    3. The attempt at a solution

    Let's assume $$f_{+}' > f_{-}'$$, as the derivative exists, it means it is $$< \infty$$.
    Therefore, $$|f(x)-f(y)|≤f_{+}'|(x-y)|$$ is a Lipschitz equation, and for
    $$|f(x)-f(x_0)|≤f_{+}'|(x-x_0)|$$
    it is a lipschitz equation in x_0.
    Thus, for the lipschitz equation properties, the function is continuos in $$x_0$$
     
    Last edited: Dec 3, 2012
  2. jcsd
  3. Dec 3, 2012 #2

    micromass

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    How exactly did you obtain these inequalitities??

    Also, you should type # instead of $. Typing

    Code (Text):

    $$ ... $$
     
    automatically places everything on a separate line. The command

    Code (Text):

    ## ... ##
     
    does not do this.
     
  4. Dec 4, 2012 #3
    [itex] ##|f(x)−f(y)|≤L|(x−y)|##[/itex]
    this is the definition of lipschitz equation,
    while
    [itex] ##|f(x)−f(x0)|≤f′+|(x−x0)|##[/itex]
    is from lagrange's theorem
     
  5. Dec 4, 2012 #4

    micromass

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    What does Lagrange's theorem say? Are you really allowed to apply it in this case? Are all the conditions satisfied?
     
  6. Dec 4, 2012 #5
    uhm.. ok, i guess i don't have all the conditions to apply lagrange actually.
    any hint about how else i can solve it?
     
  7. Dec 4, 2012 #6

    micromass

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    Did you prove already that every differentiable functions is continuous? Can you try to adapt that proof?
     
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