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Prove that a fuction is continous and differentiable everywhere, but not at f'=0

  1. Apr 28, 2012 #1
    1. The problem statement, all variables and given/known data

    Prove that the function f:ℝ→ℝ, given by

    f(x)={x2sin(1/x) if x≠0, 0 if x=0}

    is continous and differentiable everywhere, but that f' is not continuous at 0.



    2. Relevant equations



    3. The attempt at a solution

    I thought if a function was differentiable everywhere then it was continuous everywhere?

    If not what should I do??
     
    Last edited: Apr 28, 2012
  2. jcsd
  3. Apr 28, 2012 #2

    I like Serena

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    Homework Helper

    Hi charmedbeauty! :smile:

    I guess this is exactly the counter example that shows that differentiability (even everywhere) does not imply continuity of the derivative.

    What you would need to do is to check the definitions of continuity and differentiability.
    Do you have those handy?

    I'll help you along.
    A function f is continuous at point c iff ##\lim\limits_{x \to c} f(x)=f(c)##.
    Does that hold for c=0 in your case?
     
    Last edited: Apr 28, 2012
  4. Apr 28, 2012 #3

    Dick

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    Differentiability DOES imply continuity. f IS continuous. It's f' that is discontinuous.
     
  5. Apr 28, 2012 #4
    f' exists for all values of x which implies that f is differentiable and hence f is continuous too
    f' not being continuous is a different matter
     
  6. Apr 28, 2012 #5
    cos(1/x) is not continuous at x=0
     
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