Slowly Oscillating Continuity

Abstract

A function $f$ is continuous if and only if, for each point $x_0$ in the domain, ${\lim }_{n\to \infty }f\left(x_n\right)=f\left(x_0\right)$, whenever ${\lim }_{n\to \infty }{x}_n=x_0$. This is equivalent to the statement that $\left(f\left(x_n\right)\right)$ is a convergent sequence whenever $\left(x_n\right)$ is convergent. The concept of slowly oscillating continuity is defined in the sense that a function $f$ is slowly oscillating continuous if it transforms slowly oscillating sequences to slowly oscillating sequences, that is, $\left(f\left(x_n\right)\right)$ is slowly oscillating whenever $\left(x_n\right)$ is slowly oscillating. A sequence $\left(x_n\right)$ of points in $\mathbf{R}$ is slowly oscillating if ${\lim }_{\lambda \to 1^{+}}{{\bar{\lim }}_n{\max }_{n+1\le k\le [\lambda n]}}_{}\left|x_k-x_n\right|=0$, where $\left[\lambda n\right]$ denotes the integer part of $\lambda n$. Using $\varepsilon >0$'s and $\delta$'s, this is equivalent to the case when, for any given $\varepsilon >0$, there exist $\delta =\delta \left(\varepsilon \right)>0$ and $N=N\left(\varepsilon \right)$ such that if $n\ge N\left(\varepsilon \right)$ and $n\le m\le \left(1+\delta \right)n$. A new type compactness is also defined and some new results related to compactness are obtained.