Removable Singularities for Quasiregular Mappings

Given a continuous open finite-to-one mapping $ f $ of a domain $ G $ including a closed set $ E $ , for each natural $ k $ we consider the set $ E(k) $ (possibly empty) of all points in $ E $ at which $ f $ attains a value with multiplicity $ k $ over $ G $ . Suppose that each point of $ E(k) $ has a neighborhood where the restriction of $ f $ to $ E(k) $ is injective, and its inverse mapping is weakly $ (h,H) $ -quasisymmetric. If, moreover, $ f $ is quasiregular outside $ E $ , then it is quasiregular on the entire domain $ G $ . This theorem generalizes the sufficient condition for the removability of closed sets in the class of quasiconformal mappings obtained by Väisälä in 1990.