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Theorem. Let $G$ be a connected open set and let $f: G →\mathbb{C}$ be an analytic function. Then the following are equivalent statements:
(a) $f=0$;
(b) there is a point $a$ in $G$ such that $f^{(n)}(a)=0$ for each $n ≥ 0$;
(c) $\{z ∈ G:f(z)= O\}$ has a limit point in $G$.

Give an example to show that $G$ must be assumed to be connected in Theorem.

This is a problem from conway.can anyone suggest me a proper example. Thank you.

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First, do you have an example of an open set that is not connected? Do you have any thoughts on the problem? –  Jonas Meyer Dec 15 '12 at 6:39
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1 Answer 1

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Let $D$ be the open unit disk and $B$ the ball of radius $1/2$ centered at $2$. Define $f: B \cup D \rightarrow \mathbb C$ by $f|_D=0$ and $f|_B=2$.

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the function you give it has no zero.so how can i apply the theorem? –  poton Dec 15 '12 at 6:53
    
Sorry, I'm a bit tired. Try it now. –  JSchlather Dec 15 '12 at 7:01
    
thanks a lot.... –  poton Dec 15 '12 at 8:34
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