analytic functions defined on $A\cup D$

Since the zeroes of a non-zero analytic function are isolated, it may have only countably many zeroes (can you see why?).
If $f,g$ are analytic in $B\subseteq\mathbb{C}$ open, and $f(z)g(z)=0$ for all $z\in B$ then $B\subseteq N(f)\cup N(g)$ (where $N(f)$ are the zeroes of $f$). What can you say about the cardinality of $B$? could that happen if $f,g$ are both non-zero?
If $B⊆C$ is open then it is uncountable. So either $N(f)$ or $N(g)$ (or both) are uncountable. An analytic function with uncountable zeroes is zero. So either $g$ is zero or $f$ is zero, on $B$ if B is connected.
Since $A$, $D$ are connected but $A\cup D$ is not, apply the arguments to the connected components.

Hint: is $A \cup D$ connected?