On $\mathrm{\Delta }$-quasi-slowly oscillating sequences

A sequence $(x_n)$ of points in a topological group is called $\mathrm{\Delta }$-quasi-slowly oscillating if $(\mathrm{\Delta }{x}_n)$ is quasi-slowly oscillating, and is called quasi-slowly oscillating if $(\mathrm{\Delta }{x}_n)$ is slowly oscillating. A function $f$ defined on a subset of a topological group is quasi-slowly (respectively, $\mathrm{\Delta }$-quasi-slowly) oscillating continuous if it preserves quasi-slowly (respectively, $\mathrm{\Delta }$-quasi-slowly) oscillating sequences, i.e. $(f(x_n))$ is quasi-slowly (respectively, $\mathrm{\Delta }$-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.