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Math Help - Lipschitz Continuity

  1. #1
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    Lipschitz Continuity

    Hey guys, I need to do the following.

    Let c > 0 and let f(x) = e^x. Show that f(x) is Lipschitz continuous for x \in [-c,c].

    My idea is to set L = e^c, but it is hard to prove that this is a valid Lipschitz constant.
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  2. #2
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    Quote Originally Posted by EinStone View Post
    Hey guys, I need to do the following.

    Let c > 0 and let f(x) = e^x. Show that f(x) is Lipschitz continuous for x \in [-c,c].

    My idea is to set L = e^c, but it is hard to prove that this is a valid Lipschitz constant.
    Mean value theorem: f(y) - f(x) = (y-x)f'(z), for some z lying between x and y. So you can take the Lipschitz constant to be the maximum value of |f'(z)| in the given interval (which in this case is e^c, as you suspected).
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by EinStone View Post
    Hey guys, I need to do the following.

    Let c > 0 and let f(x) = e^x. Show that f(x) is Lipschitz continuous for x \in [-c,c].

    My idea is to set L = e^c, but it is hard to prove that this is a valid Lipschitz constant.
    What Opalg is just a consequence of a much broader theorem, namely that if f:E\mapsto\mathbb{R} is differentiable and f' is bounded then f is Lipschitz. (this is, in fact, an iff statement)

    And thus, if E is compact and if f is continuously differentiable we must have that f' necessarily bounded and thus f is Lipschitz.
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