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Real Analysis Question involving the definition of the derivative

  1. Nov 11, 2008 #1
    1. The problem statement, all variables and given/known data

    Suppose f:(a,b) to R(Real numbers), is differentiable at any point x that is in the interval (a,b). Prove that lim(as h goes to 0) of (f(x+h)- f(x-h))/(2h) exists and equals the derivative of f(x) ( which is f '(x)). Give an example of a function where the limit exists, but the function is not differentiable.

    2. Relevant equations

    The main equation that we can use is
    f '(x)= the limit as h goes to 0 of (f(x+h)-f(x))/h (definition of the derivative)
    Probably also need the fact that x is in the domain, and that x+h is in the domain.

    3. The attempt at a solution

    I played around with the lim (h goes to 0) (f(x+h)-(fx-h))/2h, and got
    (1/2)((f(x+h)-f(x))/h) + (1/2)((f(x)-f(x-h))/h) but I am stuck on where to go next. Any ideas would be great. Thanks
  2. jcsd
  3. Nov 11, 2008 #2


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    Homework Helper

    Change h to -h in the second difference quotient. h can go to zero positively or negatively, can't it? Aren't they both (1/2)*f'(x)?
  4. Nov 12, 2008 #3


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    Staff Emeritus
    Science Advisor

    Look at
    [tex]\frac{f(x+h)- f(x)+ f(x)- f(x-h)}{h}[/itex]
    How is that related to your problem and to the derivative?
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