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Definition of derivative

  1. Nov 25, 2008 #1
    1. The problem statement, all variables and given/known data

    I had a quick question concerning some definitions. What is the difference between lim_{x \rightarrow x_0} f'(x) to exist and for f'(x_0) to exist, definition wise?

    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Nov 25, 2008 #2
    If the function is continuous then the limit as x approaches y of f is equal to f evaluated at y. Continunity is not a required property of derivatives (there are examples to show this).
     
  4. Nov 26, 2008 #3

    HallsofIvy

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    To take a very simple example, let f(x)= x2 if x is not 1, f(1)= 2. For any x other than 1, f(x)= x2 in some interval around 1 and so it's derivative is 2x. limit as x goes to 1 of f'(x) is 2. But since f(x) is not continuous at x= 1, it is not differentiable there.
     
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