On ideal-ward compactness

Abstract

A family of sets $I\subset2^{\textbf{N}}$ is called an ideal if and only if for each $
A,B\in I,$ implies $A\cup B\in I$ and for each $A\in I$ and each $B\subset A,
$ implies $B\in I.$ A real function $f$ is ward continuous if and only if $(\Delta f(x_{n}))$ is a null sequence whenever $(x_{n})$ is a null sequence and a subset $E$ of $\textbf{R}$ is ward compact if any sequence $\textbf{x}=(x_{n})$ of points in $E$ has a quasi-Cauchy subsequence where $\textbf{R}$ is the set of real numbers. These recent known results suggest to us introducing a concept of $I$-ward continuity in the sense that a function $f$ is $I$-ward continuous if $I-\lim_{n\rightarrow\infty} \Delta f(x_{n})=0$ whenever $I-\lim_{n\rightarrow\infty} \Delta x_{n}=0$ and a concept of $I$-ward compactness in the sense that a subset $E$ of $\textbf{R}$ is $I$-ward 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 $I-\lim_{k\rightarrow \infty} \Delta z_{k}=0$ where $\Delta z_{k}=z_{k+1}-z_{k}$. We investigate $I$-ward continuity and $I$-ward compactness, and prove some related problems