If for each $x$ there is $n_x$ such that $f^{(n_x)}(x)=0,$ then $f$ is a polynomial.
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.