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Definition of differentiation

  1. May 23, 2009 #1
    1. The problem statement, all variables and given/known data

    By the formal definition of differentiation Prove that if f differentiable in c and f(c)[tex]\neq0[/tex] then |f| differentiable in c.



    3. The attempt at a solution

    I know that if f differentiable do it also continues but I stuck because this fact correct necessarily only for one direction...
     
  2. jcsd
  3. May 23, 2009 #2

    quasar987

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    Re: diffrentiability

    Let c be such that f(c) is not 0. Recall that since f is differentiable, it is continuous.

    Suppose first that f(c) >0. Intuitively, the fact that f is continuous implies that as you wobble a little around c, f wobbles a little around f(c), and so for small enough wobbling around c, f should remain of the same sign as f(c). In other words, if our understanding of continuity is correct, we should be able to prove that there exists an [itex]\epsilon>0[/itex] such that f(x)>0 for all x in [itex](c-\epsilon,c+\epsilon)[/itex]. In particular, this means that |f(c+h)|=f(c+h) for small enough h.

    And in the same way, we can show that if f(c)<0, then |f(c+h)|=-f(c+h) for small enough h.

    Hopefully, you see how you can use this to calculate the derivative of |f| at c.
     
  4. May 23, 2009 #3
    Re: diffrentiability

    Thanks,

    That was very helpful :)
     
    Last edited: May 23, 2009
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