Let $f(z,w)$ be a meromorphic function (in both $z$ and $w$) inside and on some closed contour $\gamma$. Consider
\begin{align} N(w): = \frac{1}{2\pi i} \oint \limits_{\gamma} \frac{ \frac{df(z,w)}{dz} }{f(z,w)} \mathrm{d}z \end{align} According to Argument Principle, the value of $N(w)$ is the difference between the number of zeros and poles of $f$. Since $N(w)$ is an integer number, it cannot be a continuous function of $w$. I was wondering whether $N(w)$ is semi-continuous as a function of $w$?
We can assume $w\in W$, and the function $f(z,w)$ remains meromorphic and has no zeros/poles on $\gamma$ for all the points $w\in W$.
I am new to complex analysis, I greatly appreciate any comment/response.