= Solution
For each $n\geq0$, define
$$
E_n=\{z\in\mathbb C:f^{(n)}(z)=0\}.
$$
Every $f^{(n)}$ is an <entire function>, hence continuous, so $E_n$ is closed. The hypothesis says
$$
\mathbb C=\bigcup_{n=0}^{\infty}E_n.
$$
The <complex plane> is a <complete metric space>. The <Baire category theorem> therefore implies that some $E_N$ has nonempty interior. Thus $f^{(N)}$ vanishes on an open set, and the <identity theorem> gives
$$
f^{(N)}\equiv0
$$
on all of $\mathbb C$. If $N=0$, then $f=0$; otherwise the <power series> of $f$ has no term of degree $N$ or higher. Hence $f$ is a <polynomial> of degree at most $N-1$. This is the <pointwise vanishing derivative criterion for a polynomial>.
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