On GG-continuity

A function ff on a topological space is sequentially continuous at a point uu if, given a sequence (xn)(xn), limxn=ulimxn=u implies that limf(xn)=f(u)limf(xn)=f(u). This definition was modified by Connor and Grosse-Erdmann for real functions by replacing limlim with an arbitrary linear functional GG defined on a linear subspace of the vector space of all real sequences. In this paper, we extend this definition to a topological group XX by replacing GG, a linear functional, with an arbitrary additive function defined on a subgroup of the group of all XX-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.