  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
$\overline{D}.$
