= Solution
Write $g=u+iv$ and assume its domain is open and connected. If the constant modulus is zero then $g=0$. Otherwise let $u^2+v^2=c^2>0$. Differentiating this identity and using the <Cauchy-Riemann equations> gives
$$
u u_x+v v_x=0,\qquad u u_y+v v_y=-u v_x+v u_x=0.
$$
These are a linear system for $u_x,v_x$ with coefficient <matrix> $\begin{pmatrix}u&v\\v&-u\end{pmatrix}$, whose <determinant> is $-c^2\neq0$. Therefore $u_x=v_x=0$, and the <Cauchy-Riemann equations> also give $u_y=v_y=0$. It follows that
$$
\boxed{g\text{ is constant on its connected domain}.}
$$
This proves the <constant-modulus holomorphic function> result directly. On a disconnected domain the same conclusion holds separately on each <connected component>, but the constants can differ.
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