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If $f\in C^\infty(\mathbb{R},\mathbb{R})$ is a smooth function and assume that $f$ restricted to $[a;b]$ is not a polynomial for all intervals $[a; b]\subset \mathbb{R}$ with $a < b$.

Prove that there exists $x_0 \in \mathbb{R}$ so that for all $i \in \mathbb{N}$ we have $f^{(i)}(x_0)\neq 0$. We are given a hint that we can use the Baires Category Theorem: $$ \mathbb{R} = \bigcup\limits_{k\in\mathbb{N}}A_k $$ then there exist $k_0 \in \mathbb{N}$ and interval $I = [s, t]$ with $s < t$ such that $I \subset A_{k_0}$. Please help me out!

Thank you.

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Exactly the same… – Norbert Jul 3 '12 at 10:27
Also you must say that set $A_k$ are closed. – Norbert Jul 3 '12 at 10:33

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