Prove the following proposition
If there exists a branch $F(z)$ of log$z$ in the domain $D$, then any other
branch is of the form $F(z) + 2k$$\pi i$, for some $k \in $ $\mathbf Z$
Conversely $F(z) + 2k$$\pi i$ is a branch of log$z$ for any $k \in $ $\mathbf Z$
Suppose that $F(z)$ and $G(z)$ are two branch of log$z$, then the difference
is a continuous function in $D$ which takes only integer values.
Since $D$ is a domain, in particular it is connected, such a function is necessarily constant and we will prove this fact.
The set of points $z\in D$ such that $H(z)$ is equal to a given integer $n$ is both open and closed, and so the set is either empty or equal $D$.
the constant must therefore be an integer.
Thus $F(z) + 2k$$\pi i$ is a branch of log$z$ for any integer $k$ is obvious.
Intuitively i understood what the theorem meant because the branch $H(z)$ have to
go in cycles of $2$$\pi$ and thus there will be a repetition in each cycle but
I dont quite understand the prove above especially from the part onwards
'The set of points $z\in D$ such that $H(z)$ is equal to a given integer...'
Could someone explain this proof to me. Thanks