On G-continuity

A function f on a topological space is sequentially continuous at a point u if, given a sequence (xn), lim xn = u implies that lim f (xn) = f (u). This definition was modified by Connor and Grosse-Erdmann for real functions by replacing lim with an arbitrary linear functional G defined on a linear subspace of the vector space of all real sequences. In this paper, we extend this definition to a topological group X by replacing G, a linear functional, with an arbitrary additive function defined on a subgroup of the group of all X-valued sequences and not only give new theorems in this generalized setting but also present theorems that have not been obtained for real functions so far.