Variations on Quasi Cauchy Double Sequences

Abstract

The notion of a p quasi Cauchy double sequence is introduced and investigated. A double sequence $\{x_{k,l}\}$ is called $p$-quasi-Cauchy if given an $\epsilon > 0$ there exists an $n_0 \in {\bf N}$ such that
$$\max_{r,s= 1\mbox{ and/or } 0} \left \{|x_{k,l} - x_{k+p+r,l+p+s}|\right \} < \epsilon $$ whenever $k,l > n_0 $. We study compactness types of theorems of a double subset $A\times A$ of ${\bf R}^{2}$, and continuity type properties of factorable double functions defined on a double subset $A\times A$ of ${\bf R}^{2}$ into $\textbf{R}$, and obtain interesting results related to uniform continuity, sequential continuity, continuity, compactness, and a newly introduced type of continuity of factorable double functions defined on a double subset $A\times A$ of ${\bf R}^{2}$ into $\textbf{R}$.