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

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