 Solution

An analytic continuation of $f_1$ from ${\mathcal{D}}_1$ to ${\mathcal{D}}_2$ is a holomorphic function $f_2$ on ${\mathcal{D}}_2$ whose restriction to the nonempty overlap ${\mathcal{D}}_1\cap {\mathcal{D}}_2$ agrees with $f_1$.

Suppose $f_2$ and ${\tilde{f}}_2$ are two such continuations. Both equal $f_1$ on ${\mathcal{D}}_1\cap {\mathcal{D}}_2$, soon that nonempty open set. Since ${\mathcal{D}}_2$ is connected, the identity theorem givesThus analytic continuation to ${\mathcal{D}}_2$ is unique.