product of two analytic functions is 0

Question

Let $U$ be a subset of $\mathbb{C}$, which is not connected. How can I find two analytic functions $f$ and $g$ from $U$ to $\mathbb{C}$ such that $f\neq0$ and $g\neq0$. But $f\cdot g = 0$.

thanks for any hint.

Answer 1

If $U$ is for example the disks with radius $1$ around $-2$ and $2$, then $f(z)= z $ on the first disk but 0 on the other, and vice versa for $g$ their product would be 0

Answer 2

Separate the connected domain in two parts, say, $A$ and $B$, so that $f\vert _A = 0$ but $f\vert_B=1$, and let $g$ be the opposite.

Answer 3

Let $U_1$ and $U_2$ a separation of $U$. Take $$f(z)=\begin{cases}e^z&z\in U_1\\ 0&z\in U_2\end{cases}$$ and $$g(z)=\begin{cases}e^z&z\in U_2\\ 0&z\in U_1\end{cases}.$$