  It is established a series of criteria for continuous and homeomorphic
extension to the boundary of the so-called lower $Q$-homeomorphisms $f$ between
domains in $\overline{\Rn}=\Rn\cup\{\infty\}$, $n\geqslant2$, under integral
constraints of the type $\int\Phi(Q^{n-1}(x))\,dm(x)<\infty$ with a convex
non-decreasing function $\Phi:[0,\infty]\to[0,\infty]$. It is shown that
integral conditions on the function $\Phi$ found by us are not only sufficient
but also necessary for a continuous extension of $f$ to the boundary. It is
given also applications of the obtained results to the mappings with finite
area distortion and, in particular, to finitely bi-Lipschitz mappings that are
a far reaching generalization of isometries as well as quasi-isometries in
$\Rn$. In particular, it is obtained a generalization and strengthening of the
well-known theorem by Gehring--Martio on a homeomorphic extension to boundaries
of quasiconformal mappings between QED (quasiextremal distance) domains.
