= Solution
An analytic continuation along $\alpha$ is a chain of function elements $(f_t,U_t)$ covering the path, beginning with the germ of $(f,U)$ at $\alpha(0)$, such that neighboring elements agree on the component of overlap met by the path. The terminal element is $(g,V)$.
The <Monodromy theorem> says that if continuation is possible along every path in $D$, then continuations along endpoint-fixed homotopic paths have the same terminal germ. It follows by subdividing a homotopy square into small rectangles on which the <identity theorem> identifies neighboring germs. Hence
$$
g(\alpha(1))=h(\beta(1)).
$$
The requested final example with homotopic paths and different terminal values cannot exist under these hypotheses; it contradicts the theorem just proved. With “non-homotopic” in place of “homotopic”, take the germ of $\log z$ near $1$ in $\mathbb C^*$. Continuation along the constant path gives $0$, whereas continuation once counterclockwise around zero gives $2\pi i$.
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