On $Δ$-quasi-slowly oscillating sequences

A sequence $(x_{n})$ of points in a topological group is called $Δ$-quasi-slowly oscillating if $(Δx_{n})$ is quasi-slowly oscillating, and is called quasi-slowly oscillating if $(Δx_{n})$ is slowly oscillating. A function $f$ defined on a subset of a topological group is quasi-slowly (respectively, $Δ$-quasi-slowly) oscillating continuous if it preserves quasi-slowly (respectively, $Δ$-quasi-slowly) oscillating sequences, i.e. $(f(x_{n}))$ is quasi-slowly (respectively, $Δ$-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.

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