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Solution:

Let P be the set of the poles of f suppose that it has a limit point.

p is a pole of f so it is an isolated point such that [tex]\lim_{z \rightarrow p} \mid f(z) \mid= \infty [/tex]

from isolated point definition there exist an R>0 such that f is analytic at the set B={z : 0<|z-p|<R }

but B is open and p in B, and B intersect U\{p} is not phi so f is not analytic at B

Contradiction

what about the set of zeros, and is my proof right ?