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Prove differentiability and continuity

  1. Nov 19, 2009 #1

    hth

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    1. The problem statement, all variables and given/known data

    Determine that, if f(x) =

    {xsin(1/x) if x =/= 0
    {0 if x = 0

    that f'(0) exists and f'(x) is continuous on the reals. (Sorry I can't type the function better, it's piecewise)

    2. Relevant equations

    3. The attempt at a solution

    For f'(0) existing,

    For x ≠ 0,

    (f(x) - f(0))/(x - 0) = (x sin(1/x))/x = sin(1/x).

    Sin(1/x) doesn't have a limit as x → 0 because the function oscillates between -1 and 1, f is not differentiable at x = 0. Therefore f'(0) doesn't exist.

    For f'(x) being continuous.

    For any given ɛ > 0, there exists δ > 0 such that |x - y| < δ and |f(x) - f(y)| < ɛ for all x, y on the reals.

    For all x [tex]\in[/tex] R (R is the reals), .... and this where I'm stuck.
     
  2. jcsd
  3. Nov 19, 2009 #2

    jgens

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    Gold Member

    If you know that [itex]f'(0)[/itex] doesn't exist, doesn't that prove that [itex]f'(x)[/itex] is not continuous on the reals?
     
  4. Nov 19, 2009 #3

    hth

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    Yeah, you're right. Haha, I'm sorry. Total brain letdown on that one.
     
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