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Math Help - Find where f is continuous and differentiable

  1. #1
    Junior Member mremwo's Avatar
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    Find where f is continuous and differentiable

    For all n in N (natural numbers), let f(x):= x^n if x >= (gr/eq to) 0 and f(x)=0 for x<0. For which values of n is f' continuous at 0? Which values of n is f' differentiable at 0? I can understand why this is, but I don't know how to prove it using limit definitions. Thanks for your help
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    Quote Originally Posted by mremwo View Post
    For all n in N (natural numbers), let f(x):= x^n if x >= (gr/eq to) 0 and f(x)=0 for x<0. For which values of n is f' continuous at 0? Which values of n is f' differentiable at 0? I can understand why this is, but I don't know how to prove it using limit definitions. Thanks for your help
    You need to check the limit from both sides

    The left one is easy. let \delta = \epsilon

    Then 0-x<\delta  \implies \frac{f(x)-0}{x-0} =0 < \epsilon

    Now the right limit is where this get interesting!

    x-0 < \delta  now we need to analyze

    \displaystyle \frac{x^n-0}{x-0}=x^{n-1}

    This need to equal the limit form the left! So what must the values of n be?
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