My professor says given the real part $u$ of an analytic function $f$ defined on a domain $D\subset \mathbb{C}$, that we can't rule out the possibility that there could exist some other analytic function $g$, distinct from $f$ beyond just the addition of a constant, defined on a domain $E\subset \mathbb{C}$ either disjoint from, or not homeomorphic to, $D$, provided that $f$ is not analytic on $E$.
Since differentiating $u$ with respect to one variable and then integrating it with respect to the other completely determines the imaginary part, what this says to me is that $u$ would have to either produce different partial derivatives on $D$ and $E$ respectively, or $\frac{\partial u}{\partial x}$ different primitives.
The case of D and E being disjoint is trivial, but can anyone give me an example for D and E overlapping but non-homeomorphic?