On equicontinuity of mappings with branching in the closure of a domain

In the present paper, questions about a local behavior of mappings $f:D\to \overline{{\mathbb{R}}^n},$ $n\ge 2,$ in $\bar{D}$ are studied. Under some conditions on a measurable function $Q(x),$ $Q:D\rightarrow [0, \infty],$andboundariesof$D$and$D^{,\prime}=f(D),$ it is showed that a family of open discrete map-ping $f:D\to \overline{{\mathbb{R}}^n},$ $n\ge 2,$ with characteristic of quasiconformality $Q(x),$ is equicontinuous in $\bar{D}\mathrm{.}$