= Solution
{c}
Let $\beta(\theta)=w-e^{i\theta}=|\beta(\theta)|e^{ig(\theta)}$. It is never zero, since $|w|<1$. On the <unit circle>,
$$
e^{i\theta}-w=-\beta(\theta),\qquad
1-\overline w e^{i\theta}=-e^{i\theta}\overline{\beta(\theta)}.
$$
Dividing these identities gives
$$
\boxed{f(e^{i\theta})=e^{-i\theta}\frac{\beta(\theta)}{\overline{\beta(\theta)}}
=\exp\bigl(i(-\theta+2g(\theta))\bigr).}
$$
Any other continuous lift of the argument differs by an integer multiple of $2\pi$, which leaves the exponential unchanged.
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