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.