Let $G\subset\mathbb C$ be a non-empty open connected set and the function $f:G\to \mathbb C$ be analytic. It is easy to show that if there is a point $a\in G$ such that $f^{(n)}(a)=0$ for all $n\geq 0$, then $f\equiv 0$ on $G$. (I've used Taylor's series expansion of $f$ about the point $a$)
My questions is: If the "connectedness" of $G$ is dropped/discarded from the domain of $f$, what will happen? So, my specific question is sated as follows:
Specific Question: Is there any non-zero analytic function on a non-empty open disconnected subset $G$ of $\mathbb C$ such that there is a point $a\in G$ such that $f^{(n)}(a)=0$ for all $n\geq 0$?
Thanks in advance.
Yes. Take any disconnected open set $G$ and write it as $G=U\cup V$ where $U,V$ are open, nonempty, disjoint sets. Now define $f:G\to\mathbb{C}$ by $f(z)=0$ for $z\in U$ and $f(z)=1$ for $z\in V$.