New kinds of continuities

Abstract

A sequence $\left(x_n\right)$ of points in a topological group is slowly oscillating if for any given neighborhood $U$ of $0$, there exist $\delta =\delta \left(U\right)>0$ and $N=N\left(U\right)$ such that $x_m-x_n\in U$ if $n\ge N\left(U\right)$ and $n\le m\le (1+\delta )n$. It is well known that in a first countable Hausdorff topological space, a function $f$ is continuous if and only if $\left(f\left(x_n\right)\right)$ is convergent whenever $\left(x_n\right)$ is. Applying this idea to slowly oscillating sequences one gets slowly oscillating continuity, i.e. a function $f$ defined on a subset of a topological group is slowly oscillating continuous if $\left(f\left(x_n\right)\right)$ is slowly oscillating whenever $\left(x_n\right)$ is slowly oscillating. We study the concept of slowly oscillating continuity and investigate relations with statistical continuity, lacunary statistical continuity, and some other kinds of continuities in metrizable topological groups.