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Proving a function is differetiable in R

  1. Sep 9, 2007 #1

    Simfish

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    Gold Member

    f(x) is a function with...

    [tex]| f(x) - f(y) | \leq |x-y|^2 \forall x,y \in \Re [/tex]

    (a) prove differentiability in R, find f'
    (b) prove f continuous

    ====

    my steps;

    (a) [tex]\frac{| f(x) - f(y) |}{|x-y|} \leq |x-y|[/tex]

    Then by the definition of differentiability as stated in Apostol "Mathematica Analysis pg. 104, f is differentiable if the limit of the function as x -> y exists.

    So by the inequality, as x -> y, we know that the limit is bounded and therefore must exist. The value of the limit is simply |x-y|, so is that always the derivative of the function? (or could it be |x|)? Since the derivative, after all, must always be positive?

    (b) differentiability implies continuity
     
  2. jcsd
  3. Sep 10, 2007 #2
    Using epsilon-delta definition of limit, let epsilon=delta to show the limit of f(x)-f(y)/(x-y) is zero. that is as x goes to y.

    derivatives aren't always positive. consider y=-x.

    (-1)^n n a natural number is bounded but has no limit.
     
    Last edited: Sep 10, 2007
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