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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 $C^\omega$?2012-11-22
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    http://mathoverflow.net/questions/34059/if-f-is-infinitely-differentiable-then-f-coincides-with-a-polynomial2012-11-22
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    This seems to be a duplicate of http://math.stackexchange.com/q/233112 and http://math.stackexchange.com/q/93452/2012-11-22
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    possible duplicate of [Infinitely times differentiable function](http://math.stackexchange.com/questions/233112/infinitely-times-differentiable-function)2012-12-02

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