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Simple formulation, nontrivial problem
I have the following problem: let $f : \mathbb{R} \rightarrow \mathbb{R}$ be infinitely differentiable, such that for each $x \in \mathbb{R}$ there is an $n(x) \in \mathbb{N}$ such that $f^{(n(x))}(x) = 0$. We must prove $f$ is a polynomial. Note that the difficulty of this problem lies in the fact that $n(x)$ depends on $x$. If we could find an $n$ such that $f^{(n)} = 0$ the problem would be easy.