If \(f\) is holomorphic on \(\Omega\) and not identically zero, then \(\inverseof{f}(0) \intersect \Omega\) is discrete.
Solution
It suffices to show that if \(f(a) = 0\) then \(f\) is nonzero on some \(\DD_\eps^*(a)\). Without loss of generality, suppose \(a=0\) and expand \(f(z) = \sum_{k\geq 0}c_k z^k = \sum_{k\geq m}c_k z^k\) where \(m\geq 0\) is minimal such that \(c_m\neq 0\). This exists since \(f\) is not identically zero, by uniqueness of power series. Then write \[ f(z) = \sum_{k\geq m} c_k z^k = z^m \sum_{k\geq m} c_k z^{k-m} = z^m (c_m + c_{m+1}z + \cdots) \da z^m g(z) ,\] where \(g(a) = c_m \neq 0\). Being nonzero is an open condition, so \(g\) is nonzero in some punctured neighborhood of \(a\), making \(f\) nonzero there.