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This question was asked in my complex analysis quiz and I was unable to solve it. I will ask about only options I was not able to contradict

Question: Let f be a non-constant holomorphic function in the unit disc {|z|<1} such that f(0)=1. Then it is necessary that :

(i) There are infinitely many points z in the unit disc such that |f(z)|=1 .

(ii)There are atmost finitely many points z in the unit disc such that |f(z)|=1 .

If points in (i) had a limit point then I could have used identity principle to contradict it, but I can't now and i am not able to think about any other result which can be used , so kindly help .

Thanks!!

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  • $\begingroup$ See 1 or 2 or 3 or 4. $\endgroup$
    – cqfd
    Oct 16 '20 at 16:21
  • $\begingroup$ Hint: any open neighborhood of one contains infinitely many points of modulus $1$ $\endgroup$
    – Conrad
    Oct 16 '20 at 16:45

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