$delta$-Quasi Cauchy Sequences

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.