= Solution
A complex function is <holomorphic> at a point if it is complex differentiable throughout some open neighbourhood of that point. Writing $\zeta=\xi+i\eta$ and $\psi=P+iQ$, a convenient sufficient and necessary criterion under continuously differentiable real components is the <Cauchy-Riemann equations>
$$
\boxed{P_\xi=Q_\eta,\qquad P_\eta=-Q_\xi.}
$$
They must hold throughout an open neighbourhood, rather than at an isolated point. Merely having partial <derivatives> that satisfy these equalities at one point is not enough without an appropriate differentiability hypothesis. For the composition $\psi=G\circ f$, <holomorphy> of $f$ and $G$ on the relevant open sets suffices; the additional condition $f'\neq0$ is needed when using a local inverse <conformal map>, not for <holomorphy> of the composition itself.
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