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Differentiability at x=0

  1. Nov 9, 2011 #1
    1. The problem statement, all variables and given/known data

    B(x)= xsin(1/x) when x is not equal to 0

    = 0 when x is equal to 0

    Determine if the function is differentiable at 0

    2. Relevant equations



    3. The attempt at a solution

    I get B'(x)= sin(1/x)+cos(1/x)*(-1/x) but really do not know what should be done next.. I think for B'(x) x cannot be 0, but isn't the discontinuity removed as the function is defined to be 0 at x=0? ...
     
  2. jcsd
  3. Nov 9, 2011 #2
    So it's a piecewise function right?

    What does the function have to be in order for it to be differentiable? Check with the definition of the derivative.
     
  4. Nov 9, 2011 #3
    I think the requirements are: 1)the derivative exists at a point 2)limits approaching from both sides of that point are the same ?

    So, the derivative does not exist at 0, BUT isn't it defined at 0 ? Does that mean the derivative actually exists and the function can be differentiate?
     
  5. Nov 9, 2011 #4

    Dick

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    Science Advisor
    Homework Helper

    Write the derivative as a difference quotient. f'(0) should be lim h->0 (f(h)-f(0))/h. Pick a specific sequence approaching 0, say h_n=1/(pi*n/2) for n an integer. So h_n->0 as n->infinity. Is there a limit? It's actually pretty helpful to sketch a graph of the function.
     
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