$delta$-Quasi Cauchy Sequences

TL;DR

This paper introduces and studies the concepts of second forward continuity and second forward compactness, extending existing ideas of forward continuity and compactness by involving second differences, and explores their properties and implications.

Contribution

It defines second forward continuity and second forward compactness, analyzing their properties and how they relate to existing notions of sequence convergence and set compactness.

Findings

Established the definitions of second forward continuity and compactness.

Proved theorems relating second forward concepts to classical notions.

Explored the impact of different convergence definitions on these properties.

Abstract

Recently, a concept of forward continuity and a concept of forward compactness are introduced in the senses that a function $f$ is forward continuous if $\lim_{n\to\infty} \Delta f(x_{n})=0$ whenever $\lim_{n\to\infty} \Delta x_{n}=0$,\; and a subset $E$ of $\textbf{R}$ is forward compact if any sequence $\textbf{x}=(x_{n})$ of points in $E$ has a subsequence $\textbf{z}=(z_{k})=(x_{n_{k}})$ of the sequence $\textbf{x}$ such that $\lim_{k\to \infty} \Delta z_{k}=0$ where $\Delta z_{k}=z_{k+1}-z_{k}$. These concepts suggest us to introduce a concept of second forward continuity in the sense that a function $f$ is second forward continuous if $\lim_{n\to\infty}\Delta^{2}f(x_{n})=0$ whenever $\lim_{n\to\infty}\Delta^{2}x_{n}=0$, and a subset $E$ of $\textbf{R}$ is second forward compact if whenever $\textbf{x}=(x_{n})$ is a sequence of points in $E$ there is a subsequence $\textbf{z}=(z_{k})=(x_{n_{k}})$ of $\textbf{x}$ with $\lim_{k\to \infty} \Delta^{2}z_{k}=0$ where $\Delta^{2} y_{n}=y_{n+2}-2y_{n+1}+y_{n}$. We investigate the impact of changing the definition of convergence of sequences on the structure of forward continuity in the sense of second forward continuity, and compactness of sets in the sense of second forward compactness, and prove related theorems.