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Thread: Need some help in showing that the function f '(x) is not continuous at 0.

  1. #1
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    Need some help in showing that the function f '(x) is not continuous at 0.

    Here is the problem: Consider the function f(x) defined by
    f(x)= x2sin(1/x) if x does not = 0
    0 if x = 0

    Show that the function f ' (x) is not continuous at 0.

    Any help would be appreciated.
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    Re: Need some help in showing that the function f '(x) is not continuous at 0.

    Quote Originally Posted by marsalanuddin View Post
    Here is the problem: Consider the function f(x) defined by
    f(x)= x2sin(1/x) if x does not = 0
    0 if x = 0

    Show that the function f ' (x) is not continuous at 0.

    Any help would be appreciated.
    Use the limit definition of the derivative and the squeeze theorem to prove that $\displaystyle f(x)$ is differentiable. Then just use the product and chain rule to find the derivative. Once you have $\displaystyle f'(x)$ show that $\displaystyle \lim_{x \to 0}f'(x) \ne f'(0)$

    Hint the limit of the derivative does not exist
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