Let $f$ be a meromorphic function in a domain $D$. The set of zeros $Z_f$ and the set of poles $P_f$ are both discrete in $D$; it means that doesn't exist a sequence of zeros (risp. sequence of poles) that converges to a zero of $f$ (risp. to a pole of $f$). My question is the following:
Let $a\in D$ such that $a\not\in Z_f $ and $a\notin P_f$; does exist a sequence $\{b_n\}$ with $b_n\in Z_f\cup P_f$ such that $b_n\rightarrow a$? Roughly speaking, can $a$ be an accumulation point for $Z_f\cup P_f$?
I've tried to give myself an answer but I don't know if it is correct:
If $f$ is continuous then $\lim_{b_n\to a} f(b_n)=f\big(\lim_{b_n\to a}b_n\big)$ but in the above case $\lim_{b_n\to a}f(b_n)=0,\infty$ or it doesn't exist, and $f\big(\lim_{b_n\to a}b_n\big)=f(a)$ Contradiction!