Consider the class of meromorphic functions $\{f_n\}_{n\in\mathbb{N}}$ given explicitly by
\begin{equation} f_n(z)=\frac{1}{n}\sum_{a=1}^{n}\frac{1}{z-a/n}\ . \end{equation}
For finite $n$, this function is a globally defined function with $n$ simple poles. However, if I take the limit $n\to\infty$ this sum reduces to a Riemann integral, and we have
\begin{equation} f(z)\equiv\lim_{n\to\infty}f_n(z)=\int_{0}^{1}\frac{\mathrm{d}u}{z-u}=\log\frac{z}{z-1}\ , \end{equation}
which has a branch cut between $z=0$ and $z=1$, and a the composition $f\circ\gamma$ of any closed path $\gamma:I\to\mathbb{C}$ encircling $z=0$ or $z=1$ picks up a monodromy, namely $f(\gamma(1))-f(\gamma(0))=2\pi n$ for some $n$.
How does this work? We start with a class of functions all of which have vanishing monodromy around any point, and end up with a function only globally defined on an infinitely-sheeted Riemann surface. I kind of see how infinitely many poles can "coalesce" into a branch cut, but the origin of the monodromy is surprising. Also, is there a general way to find what kind of branch cut will pop out of a limit like this? Could I take a limit of globally defined functions and end up with a limit that has a $\sqrt{z(z-1)}$ branch cut?