Methods of continuation of holomorphic functions of several complex variables are investigated within the axiomatic framework of Araki, Haag, and Kastler in local quantum field theory. The motivation comes from the analysis of a mass gap in an energy-momentum spectrum without vacuum vector. The main conclusion is some non-restrictedness property in a mass gap situation. Prior to that, some results on holomorphic functions related to a mass-gap-situation are obtained and investigated.