Analytic function and connected region. We have the result.
Let $G$ be an open connected set in $\mathbb{C}$, and let $f : G \rightarrow \mathbb{C}$ be an analytic function. Then the following statements are equivalent.
*1. $f(z) = 0$ in $G$ for all $z$ *
2. There is a point $a \in G$ s.t. $f^n(a) = 0$ for each $n \in \mathbb{Z}$.
3. $\{z \in G : f(z) = 0\}$ has a limit point in $G$
Can we say $\mathbb{R}$ is open connected set in $\mathbb{C}$? Why or why not?
If we do not consider $G$ as a open connected region, the result will loss its validity. Please give me a counterexample of a function for which it may happen.
Thank you for your kind suggestions.

$\mathbb{R}$ is not an open set in $\mathbb{C}$. If the set is not connected then you can  take the value of the function to be  0 on one connected component and any other constant in another component.  
