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Math Help - continuous function proof

  1. #1
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    continuous function proof

    Suppose that f(x) is any function that is continuous at x=0.
    Prove that the function g(x) = xf(x) is differentiable at x=0.

    Now my first instinct was to say that this cannot be done.
    If a function is continuous at a certain value of x does not mean its differentiable at that point as well.

    Can someone shed some light here.

    Maybe the question intends for the function to be differentiable at (x) as well.
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  2. #2
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    Opalg's Avatar
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    Go back to the definition: g(x) is differentiable at 0 if \lim_{h\to0}\frac{g(h)-g(0)}h exists. So, does that limit exist, or not?
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  3. #3
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    colby2152's Avatar
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    Maybe the question intends for the function to be differentiable at (x) as well.
    Probably, because continuity doesn't imply differentiability. However, if you can take a derivative at a point, then it does mean the function is continuous at that point.
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  4. #4
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    it could, depending on the function

    |x| for example
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