Functions preserving slowly oscillating double sequences

TL;DR

This paper investigates the continuity properties of functions defined on double sequences that are slowly oscillating, introducing new types of continuity and analyzing their implications in the context of real-valued functions.

Contribution

It introduces and studies new continuity concepts for factorable double functions on slowly oscillating double sequences, expanding understanding of their behavior.

Findings

Characterization of uniform and sequential continuity for these functions

Introduction of a new type of continuity for factorable double functions

Results linking oscillation conditions to continuity properties

Abstract

A double sequence $\textbf{x}=\{x_{k,l}\}$ of points in $\textbf{R}$ is slowly oscillating if for any given $\varepsilon>0$, there exist $\alpha=\alpha(\varepsilon)>0$, $\delta=\delta (\varepsilon) >0$, and $N=N(\varepsilon)$ such that $|x_{k,l}-x_{s,t}|<\varepsilon$ whenever $k,l\geq N(\varepsilon)$ and $k\leq s \leq (1+\alpha)k$, $l\leq t \leq (1+\delta)l$. We study continuity type properties of factorable double functions defined on a double subset $A\times A$ of $\textbf{R}^{2}$ into $\textbf{R}$, and obtain interesting results related to uniform continuity, sequential continuity, and a newly introduced type of continuity of factorable double functions defined on a double subset $A\times A$ of $\textbf{R}^{2}$ into $\textbf{R}$.