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Homework Help: Analytic function

  1. Dec 13, 2008 #1
    1. The problem statement, all variables and given/known data
    Let f:ℝ→ℝ an analytic function,i.e, f ∈ C^{∞}(ℝ) and for all a∈ℝ we have that
    f:∑[tex]\infty[/tex]0((f⁽ⁿ⁾(a))/(n!))(x-a)ⁿ.
    Suppose that for all x in ℝ:∃N=N_{x}∈ℕ:f^{(N)}(x)=0.
    Show that F is a polynomial



    2. Relevant equations



    3. The attempt at a solution

    I think that maybe the theorem of Baire's Categories is the key of exercise. So, I said:
    Let A_{n}={x∈ℝ:f⁽ⁿ⁾(x)=0} and I want to show that for some n and for a uncountable x∈ℝ f⁽ⁿ⁾(x)=0...But I don't know how....
     
    Last edited: Dec 13, 2008
  2. jcsd
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