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.