## Abstract

A function $f$ is continuous if and only if, for each point ${x}_{0}$ in the domain, ${\mathrm{lim}}_{n\to \infty}f\left({x}_{n}\right)=f\left({x}_{0}\right)$, whenever ${\mathrm{lim}}_{n\to \infty}{x}_{n}={x}_{0}$. 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 ${\mathrm{lim}}_{\lambda \to {1}^{+}}{{\stackrel{\u2015}{\mathrm{lim}}}_{n}{\mathrm{max}}_{n+1\le k\le [\lambda n]}}_{}\left|{x}_{k}-{x}_{n}\right|=0$, 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.

## Citation

H. Çakalli. "Slowly Oscillating Continuity." Abstr. Appl. Anal. 2008 1 - 5, 2008. https://doi.org/10.1155/2008/485706

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