= Solution
Write $g=u+iv$ and let its constant <modulus> be $C$. If $C=0$, then $g=0$. For $C>0$, differentiating $u^2+v^2=C^2$ with respect to $x$ and $y$, then applying the <Cauchy-Riemann equations>, gives
$$
u u_x+v v_x=0,\qquad u u_y+v v_y=-u v_x+v u_x=0.
$$
Thus
$$
\begin{pmatrix}u&v\\v&-u\end{pmatrix}\begin{pmatrix}u_x\\v_x\end{pmatrix}=0.
$$
The coefficient <matrix> has <determinant> $-(u^2+v^2)=-C^2\ne0$, forcing $u_x=v_x=0$. The <Cauchy-Riemann equations> then also give $u_y=v_y=0$. Therefore $u,v$ are constant on a connected domain and
$$
\boxed{g\text{ is constant}.}
$$
This proves the <constant-modulus holomorphic function> result. If a domain is disconnected, the conclusion applies separately to each connected component.
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