1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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:

    and left derivative
    in $$x_0\in A$$, then $$f$$
    is continuos in

    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
    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
    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,
    [itex] ##|f(x)−f(x0)|≤f′+|(x−x0)|##[/itex]
    is from lagrange's theorem
  5. Dec 4, 2012 #4
    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
    Did you prove already that every differentiable functions is continuous? Can you try to adapt that proof?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook