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    Differentiable and continuous functions

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    Suppose fâ?¶Râ?'R is twice differentiable with both f' and f'' continuous in an interval around 0. Suppose further that f(0)=0. Let
    h(x)={f(x)/x, if xâ? 0,
    f^' (0), if x=0.
    Show that
    (a) h is differentiable at x=0.
    (b) h is differentiable at x=0 with h^' (0)=1/2 f^'' (0).
    (c) h' is continuous at x=0.

    © BrainMass Inc. brainmass.com December 24, 2021, 9:35 pm ad1c9bdddf
    https://brainmass.com/math/graphs-and-functions/differentiable-continuous-functions-395200

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    Proof:
    Since is twice differentiable function, and are continuous in the neighborhood of 0, and , then we have
    , where is in the neighborhood of 0.
    Then we get

    Since and are continuous, we have

    Now we consider if and .
    (a) First, I claim that is differentiable at . We have

    Thus is differentiable at .
    (b) From (a), we get
    (c) Next, I claim that is continuous at
    For , we have , then . Then we have

    Let , then , then we have

    Therefore, is continuous at .

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 9:35 pm ad1c9bdddf>
    https://brainmass.com/math/graphs-and-functions/differentiable-continuous-functions-395200

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