Simply $sm$-factorizable (para)topological groups and their completions

Let us call a (para)topological group emph{strongly submetrizable} if it admits a coarser separable metrizable (para)topological group topology. We present a characterization of simply $sm$-factorizable (para)topo-logical groups by means of continuous real-valued functions. We show that a (para)topo-logical group $G$ is a simply $sm$-factorizable if and only if for each continuous function $fcolon Gto mathbb{R}$, one can find a continuous homomorphism $varphi$ of $G$ onto a strongly submetrizable (para)topological group $H$ and a continuous function $gcolon Hto mathbb{R}$ such that $f=gcircvarphi$. This characterization is applied for the study of completions of simply $sm$-factorizable topological groups. We prove that the equalities $mu{G}=varrho_omega{G}=upsilon{G}$ hold for each Hausdorff simply $sm$-factorizable topological group $G$. This result gives a positive answer to a question posed by Arhangelskii and Tkachenko in 2018. Also, we consider realcompactifications of simply $sm$-factorizable paratopological groups. It is proved, among other results, that the realcompactification, $upsilon{G}$, and the Dieudonne completion, $mu{G}$, of a regular simply $sm$-factorizable paratopological group $G$ coincide and that $upsilon{G}$ admits the natural structure of paratopological group containing $G$ as a dense subgroup and, furthermore, $upsilon{G}$ is also simply $sm$-factorizable. Some results in [emph{Completions of paratopological groups, Monatsh. Math. textbf{183} (2017), 699--721}] are improved or generalized.