I am solving assignment questions of complex analysis which might not be discussed due to pendamic and I was unable to solve this particular question.
Let f be an entire function that has uncountably many zeroes . Prove that it is constant.
I was unable to get an idea on which result to use although I am good in complex analysis. So, It's my humble request to you tp kindly guide me.
I thought of $\prod_{r_i varies in\in \mathbb{R}} (z-r_i) $ , and I don't know why it must be constant.
On
Analytic functions which are non-constant are such that their zeroes are isolated, that is, for every zero $z_0$ of $f$ there is a neighborhood $U$ about $z_0$ where $f(z)\not=0$ except if $z=z_0$. We say that the set of zeroes is totally disconnected.
It remains to show that an uncountable set cannot be totally disconnected.
Hint: If for each $z\in S\subset\Bbb C$ we find $r_z>0$ such that the open ball $B_{r_z}(z)$contains no point of $S$ other than $z$, then the balls $B_{r_z/2}(z)$ are pairwise disjoint and each contains a point with rational coordinates.