Prove: $f: \mathbb{R} \rightarrow \mathbb{R}$ st for every $x \in \mathbb{R}$ there exists $n$ st $f^{(n)}(x) = 0$, f is a polynomial.

Question

If $f: \mathbb{R} \rightarrow \mathbb{R}$ is a smooth function such that for every $x \in \mathbb{R}$ there exists $n$ such that $f^{(n)}(x) = 0$, then f is a polynomial.

I'm kind of lost on this one. I know that I have to use Baire's category theorem somewhere here, but I'm not sure exactly how.

Answer 1

This problem can indeed be solved using the Baire Category Theorem.

A nice discussion of variants of the problem, along with an outline of a proof of your version, can be found in the answers to this post at MathOverflow.