  The manuscript is devoted to the boundary behavior of mappings with bounded
and finite distortion, which has been actively studied recently. We consider
mappings of domains of the Euclidean space that satisfy the inverse Poletsky
inequality with an integrable majorant, are open, and discrete. Assume that the
image of the boundary of the original domain is finitely connected relative to
the mapped domain, and the preimage of the boundary of the latter is nowhere a
dense set. Then, under certain conditions on the geometry of these domains, it
is proved that the specified mappings have a continuous boundary extension. The
result is valid even in a more general form, when the majorant in the inverse
Poletsky inequality is integrable over almost all concentric spheres centered
at each point. In particular, the obtained results are valid for homeomorphisms
as well as for open discrete closed mappings with the appropriate modulus
condition.
