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Smooth Functions

  1. Sep 25, 2008 #1
    My lecturer gave me a question that included giving a proof that a particular function is smooth. I have taken a course on analysis and have no problems when it comes to proof of continuity; i was just wondering what the usual steps are in proving that a function is smooth.
    I would guess that it would involve some sort of induction on the derivatives but if someone could sketch out a general proof that would be grand.
  2. jcsd
  3. Sep 25, 2008 #2


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    There are some rules like: sums, products and compositions of smooth functions are smooth. Usually one uses these to show that a function is smooth. Otherwise, induction on the derivatives might work.

    Since it is only a small portion of your entire exercise, could you post the function?
  4. Sep 25, 2008 #3
    Yes well this specific question is stated as such :
    f(x) is a smooth function, prove the function

    G(x) = f'(0) , x = 0
    (f(x) - f(0))/x , otherwise

    is smooth.
    I previously assumed by product rule it is true that G(x) is smooth when x is not equal to zero but obviously the whole point of the question is about x=0.
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