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 obtain theorems which are not appeared even for real functions so far.