H. Çakalli

Copyright © 2008 H. Çakalli. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A function f is continuous if and only if, for each point x0 in the domain, lim⁡n→∞f(xn)=f(x0), whenever lim⁡n→∞xn=x0. This is equivalent to the statement that (f(xn)) is a convergent sequence whenever (xn) 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(xn)) is slowly oscillating whenever (xn) is slowly oscillating. A sequence (xn) of points in R is slowly oscillating if limλ→1+lim―nmaxn+1≤k≤[λn]|xk-xn|=0, where [λn] denotes the integer part of λn. Using ɛ>0's and δ's, this is equivalent to the case when, for any given ɛ>0, there exist δ=δ(ɛ)>0 and N=N(ɛ) such that |xm−xn|<ɛ if n≥N(ɛ) and n≤m≤(1+δ)n. A new type compactness is also defined and some new results related to compactness are obtained.