On ΔΔ-quasi-slowly oscillating sequences

A sequence (xn)(xn) of points in a topological group is called ΔΔ-quasi-slowly oscillating if (Δxn)(Δxn) is quasi-slowly oscillating, and is called quasi-slowly oscillating if (Δxn)(Δxn) is slowly oscillating. A function ff defined on a subset of a topological group is quasi-slowly (respectively, ΔΔ-quasi-slowly) oscillating continuous if it preserves quasi-slowly (respectively, ΔΔ-quasi-slowly) oscillating sequences, i.e. (f(xn))(f(xn)) is quasi-slowly (respectively, ΔΔ-quasi-slowly) oscillating whenever (xn)(xn) 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.