= Solution
Define $F(z)=\overline{f(\overline z)}$. To apply the <identity theorem>, we first verify that $F$ is <holomorphic>; mere <complex conjugation> of the values of $f$ would not suffice. For $h\ne0$,
$$
\frac{F(z+h)-F(z)}h
=\overline{\frac{f(\overline z+\overline h)-f(\overline z)}{\overline h}}
\longrightarrow\overline{f'(\overline z)}.
$$
Thus $F$ is an <entire function>, just as $f$ is. At each positive integer $n$, the point $1/n$ is real and $f(1/n)$ is real, so $F(1/n)=f(1/n)$. These distinct points accumulate at $0\in\mathbb C$. The <identity theorem> on the <connected> <complex plane> gives
$$
\boxed{f(z)=\overline{f(\overline z)}\quad(z\in\mathbb C)}.
$$
This is <conjugate reflection of a holomorphic function>, with the reality data forcing equality to its reflection.
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