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Let $f$, $g$ be analytic function defined on $A\cup D$ where $A = \{z \in \mathbb{C}: \frac{1}{2}<|z|<1\}$ and $D = \{z \in \mathbb{C}: |z-2|<1\}$ Which of the following statements are true?

  1. If $f(z) g(z) =0$ for all $z \in A\cup D$, then either $f(z)=0$ for all $z \in A$ or $g(z) =0$ for all $z \in A$.
  2. If $f(z) g(z) =0$ for all $z \in D$, then either $f(z)=0$ for all $z \in D$ or $g(z) =0$ for all $z \in D$.
  3. If $f(z) g(z) =0$ for all $z \in A$, then either $f(z)=0$ for all $z \in A$ or $g(z) =0$ for all $z \in A$.
  4. If $f(z) g(z) =0$ for all $z \in A\cup D$, then either $f(z)=0$ for all $z \in A\cup D$ or $g(z) =0$ for all $z \in A\cup D$.

I am stuck on this problem. Can anyone help me please? where should I begin......................

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    For some basic information about writing math at this site see e.g. [here](http://meta.math.stackexchange.com/questions/5020/), [here](http://meta.stackexchange.com/a/70559/155238), [here](http://meta.math.stackexchange.com/questions/1773/) and [here](http://math.stackexchange.com/editing-help#latex). I tried to improve your post using TeX (for better readability). Please check whether these edits did not unintentionally change the meaning of your post.2012-12-18
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    How many zeroes can an analytic non-zero function have?2012-12-18
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    But could it be uncountable?2012-12-18
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    @pankaj: I've merged the identical version of this question which you asked 5 hours ago into this one. For future reference, **don't do that**. Posting duplicate copies of the same question is rather frowned upon on this website.2012-12-18

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