= Solution
The <identity theorem> can be expressed as the principle of isolated zeros: a <holomorphic function> on a connected open set is either identically zero or has isolated zeros. Locally this follows from its <Taylor series>: at a zero of a function which is not locally zero, factor the first nonzero power, $f(z)=(z-z_0)^m h(z)$ with $h(z_0)\ne0$. Connectedness then rules out a nonempty open region of zeros unless the whole function vanishes.
Set $g(z)=\overline{f(\bar z)}$. This is <holomorphic> on the punctured plane: reflecting and conjugating a local power series produces another power series. The absence of an <essential singularity> means that the singularity at zero is removable or a pole. Thus for some integer $m\geq0$, $z^m f(z)$ extends holomorphically across zero; the same is true of $z^m g(z)$. Their difference
$$
H(z)=z^m(f(z)-g(z))
$$
therefore extends to a <holomorphic function> on all of $\mathbb C$. At every $1/n$, real-valuedness gives $H(1/n)=0$. These zeros accumulate at zero, which is now an interior point of the domain of $H$. The <identity theorem> forces $H\equiv0$. For $z\ne0$ we may divide by $z^m$, yielding
$$
\boxed{f(z)=\overline{f(\bar z)}.}
$$
The conjugation outside $f$ is essential. Also, applying the <identity theorem> directly on the punctured plane would be invalid: the accumulation point zero does not belong to that domain.
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