On G-Continuity

TL;DR

This paper generalizes the concept of G-continuity from real functions to functions on topological groups, introducing new theorems and extending existing notions of sequential continuity.

Contribution

It extends G-continuity to topological groups using additive functions on subgroups, providing novel theorems not previously available for real functions.

Findings

Extended G-continuity to topological groups.

Proved new theorems in the generalized setting.

Discovered theorems not known for real functions.

Abstract

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.