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What's the definition of a point being differentiable?

  1. Nov 30, 2005 #1
    "Suppose f is continuous on [a,b] and c in (a,b). Suppose f is differentiable at all points of (a,b) except possibly at c. Assume further that lim(x->c)f'(x) exists and is equal to k. Prove that f is differentiable at c and f'(c)=k"

    Since the lim f'(x) as x->c exists, f'(c) either equals k, exists but doesn't equal to k, or undefined. I showed that if it is defined, it must equal to k by using the intermediate value property of f'. But I can't show that f'(c) has to be defined. I tried contradiction, saying if f'(c) is undefined, but I can't run into a contradiction.
     
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  3. Nov 30, 2005 #2

    Zurtex

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    What's the definition of a point being differentiable?
     
  4. Nov 30, 2005 #3
    lim(x->c) (f(x)-f(c))/(x-c) = f'(c), provided the limit exists.
     
  5. Nov 30, 2005 #4

    HallsofIvy

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    Try using the mean value theorem on (x0, c) and (c, x1) to show that the left and right limits of the difference quotient exist and are the same.
     
  6. Nov 30, 2005 #5
    I'm not sure what you mean by difference quotient. The mean value theorem assumes the existence of f'(c).
     
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