= Solution
The domain has two <connected components>. The zeros force the function to vanish on the <upper half-plane> but impose nothing on the lower one. Define
$$
\boxed{f(z)=\begin{cases}0,&\operatorname{Im}z>0,\\1,&\operatorname{Im}z<0.\end{cases}}
$$
This function is locally constant and hence <holomorphic> on the stated <open set>, and is not identically zero on that whole set. Thus the answer is \b[yes]. <Connectedness> is exactly the missing hypothesis for a global identity conclusion.
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