On $\Delta$-quasi-slowly oscillating sequences

TL;DR

This paper introduces and studies $ riangle$-quasi-slowly oscillating sequences and functions in topological groups, exploring their properties and relationships with various types of statistical and lacunary statistical continuities.

Contribution

It defines $ riangle$-quasi-slowly oscillating sequences and functions, and investigates their properties and connections with other continuity concepts in metrizable topological groups.

Findings

$ riangle$-quasi-slowly oscillating sequences are characterized in topological groups.

Relations between $ riangle$-quasi-slowly oscillating continuity and statistical continuity are established.

The study reveals new links between different types of oscillation-based continuities.

Abstract

A sequence $(x_{n})$ of points in a topological group is called $\Delta$-quasi-slowly oscillating if $(\Delta x_{n})$ is quasi-slowly oscillating, and is called quasi-slowly oscillating if $(\Delta x_{n})$ is slowly oscillating. A function $f$ defined on a subset of a topological group is quasi-slowly (respectively, $\Delta$-quasi-slowly) oscillating continuous if it preserves quasi-slowly (respectively, $\Delta$-quasi-slowly) oscillating sequences, i.e. $(f(x_{n}))$ is quasi-slowly (respectively, $\Delta$-quasi-slowly) oscillating whenever $(x_{n})$ is. We study these kinds of continuities, and investigate relations with statistical continuity, lacunary statistical continuity, and some other types of continuities in metrizable topological groups.