New kinds of continuities

A sequence (xn) of points in a topological group is slowly oscillating if for any given neighborhood U of 0, there exist δ = δ(U) > 0 and N = N(U) such that xm−xn ∈ U if n ≥ N(U) and n ≤ m ≤ (1+δ)n. It is well known that in a first countable Hausdorff topological space, a function f is continuous if and only if (f (xn)) is convergent whenever (xn) is. Applying this idea to slowly oscillating sequences one gets slowly oscillating continuity, i.e. a function f defined on a subset of a topological group is slowly oscillating continuous if (f (xn)) is slowly oscillating whenever (xn) 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.