New kinds of continuities

New kinds of continuities

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Article ID: iaor20113223
Volume: 61
Issue: 4
Start Page Number: 960
End Page Number: 965
Publication Date: Feb 2011
Journal: Computers and Mathematics with Applications
Authors:
Keywords: topology
Abstract:

A sequence ( x n ) equ1 of points in a topological group is slowly oscillating if for any given neighborhood U equ2 of 0 equ3, there exist δ = δ ( U ) > 0 equ4 and N = N ( U ) equ5 such that x m x n U equ6 if n = N ( U ) equ7 and n = m = ( 1 + d ) n equ8. It is well known that in a first countable Hausdorff topological space, a function f equ9 is continuous if and only if ( f ( x n ) ) equ10 is convergent whenever ( x n ) equ11 is. Applying this idea to slowly oscillating sequences one gets slowly oscillating continuity, i.e. a function f equ12 defined on a subset of a topological group is slowly oscillating continuous if ( f ( x n ) ) equ13 is slowly oscillating whenever ( x n ) equ14 is slowly oscillating. We study the concept of slowly oscillating continuity and investigate relations with statistical continuity, lacunary statistical continuity, and some other kinds of continuities in metrizable topological groups.

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