# Analytic function

1. Dec 13, 2008

### takeuchi

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:∑$$\infty$$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