Pointwise vanishing derivative criterion for a polynomial
= Pointwise vanishing derivative criterion for a polynomial
Suppose that for every $z\in\mathbb C$, at least one derivative $f^{(n)}(z)$ of an <entire function> vanishes. The closed sets $E_n=\{z:f^{(n)}(z)=0\}$ cover $\mathbb C$. By the <Baire category theorem>, some $E_N$ has nonempty interior. The <identity theorem> gives $f^{(N)}=0$ everywhere, so $f$ is a <polynomial> of degree below $N$.