Richard F. Patterson | Searcharxiv

arXiv2013

Functions preserving slowly oscillating double sequences

A double sequence $\textbf{x}=\{x_{k,l}\}$ of points in $\textbf{R}$ is slowly oscillating if for any given $\varepsilon>0$, there exist $α=α(\varepsilon)>0$, $δ=δ(\varepsilon) >0$, and $N=N(\varepsilon)$ such that $|x_{k,l}-x_{s,t}|<\varepsilon$ whenever $k,l\geq N(\varepsilon)$ and $k\leq s \leq (1+α)k$, $l\leq t \leq (1+δ)l$. 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 $\textbf{R}$.

math.GM↗

arXiv2013

On Double Sequences

A double sequence $\{x_{k,l}\}$ is quasi-Cauchy if given an $ε> 0$ there exists an $N \in {\bf N}$ such that $$\max_{r,s= 1\mbox{ and/or} 0} \left \{|x_{k,l} - x_{k+r,l+s}|< ε\right \} .$$ We study 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, and a newly introduced type of continuity of factorable double functions defined on a double subset $A\times A$ of ${\bf R}^{2}$ into $\textbf{R}$.

math.GM↗