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