On Delta-quasi-slowly oscillating sequences

A sequence (x(n)) of points in a topological group is called Delta-quasi-slowly oscillating if (Delta x(n)) is quasi-slowly oscillating, and is called quasi-slowly oscillating if (Delta x(n)) is slowly oscillating. A function f defined on a subset of a topological group is quasi-slowly (respectively, Delta-quasi-slowly) oscillating continuous if it preserves quasislowly (respectively, Delta-quasi-slowly) oscillating sequences, i.e. (f (x(n))) is quasi-slowly (respectively, Delta-quasi-slowly) oscillating whenever (x(n)) is. We study these kinds of continuities, and investigate relations with statistical continuity, lacunary statistical continuity, and some other types of continuities in metrizable topological groups. (C) 2011 Elsevier Ltd. All rights reserved.