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Let $ f: \Bbb R \to \Bbb R$ be $C^{\infty}$. Such that for each $x\in \Bbb R$ there exist a natural number $n=n_x $ such that $ f^{(n)}(x)=0$. Let's consider the set $$ J = \left\{ x:\exists \left( {a,b} \right) \ni x\,;\,f|_{\left( {a,b} \right)} \text{is a polynomial} \right\} $$ Prove that the complement $ F = J^c$ has no intervals $[a,b]$ or in other words, F has empty interior, or in other words, J is dense.

I think that I have to use Taylor expansion but I don't know how :S

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Taylor expansion? $C^\infty$ or $C^\omega$? –  Hagen von Eitzen Nov 22 '12 at 21:28
2  
    
This seems to be a duplicate of math.stackexchange.com/q/233112 and math.stackexchange.com/q/93452 –  user50376 Nov 22 '12 at 22:49
    
possible duplicate of Infinitely times differentiable function –  Lukas Geyer Dec 2 '12 at 6:13

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