Zeros of Generalized Holomorphic Functions

Let f be a generalized holomorphic function on a connected open set W ì \Bbb C\Omega\subset {\Bbb C} . It is proved that f equals zero if and only if there exists a smooth curve and a set A of positive (one-dimensional) measure such that f takes zero value on A. Also, a holomorphic generalized function different from zero on the disc, which takes zero values on a dense G _δ-set of the disc, is constructed. The generalized zero set of a holomorphic function is introduced and studied in an analogous way.