On G-continuity

A function f on a topological space is sequentially continuous at a point u if, given a sequence (x(n)), lim x(n) = u implies that lim f (x(n)) 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. (C) 2010 Elsevier Ltd. All rights reserved.