Slowly Oscillating Continuity

Abstract

A function $f$ is continuous if and only if, for each point ${x}_{0}$ in the domain, ${\lim{}}_{n{\rightarrow}\infty{}}f({x}_{n})=f({x}_{0})$, whenever ${\lim{}}_{n{\rightarrow}\infty{}}{x}_{n}={x}_{0}$. This is equivalent to the statement that $(f({x}_{n}))$ is a convergent sequence whenever $({x}_{n})$ 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, $(f({x}_{n}))$ is slowly oscillating whenever $({x}_{n})$ is slowly oscillating. A sequence $({x}_{n})$ of points in $\mathbf{R}$ is slowly oscillating if ${\lim{}}_{\lambda{}{\rightarrow}{1}^{+}}{{\stackrel{\rule{10pt}{1pt}}{\lim{}}}_{n}{\max{}}_{n+1\leq{}k\leq{}[\lambda{}n]}}_{}|{x}_{k}-{x}_{n}|=0$, where $[\lambda{}n]$ 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 $\delta{}=\delta{}(\varepsilon{})>0$ and $N=N(\varepsilon{})$ such that $|{x}_{m}-{x}_{n}|< \varepsilon{}$ if $n\geq{}N(\varepsilon{})$ and $n\leq{}m\leq{}(1+\delta{})n$. A new type compactness is also defined and some new results related to compactness are obtained.