  We introduce a certain integrability condition for the reciprocal of the
Jacobian determinant which guarantees the local homeomorphism property of
quasiregular mappings with a small inner dilatation. This condition turns out
to be sharp in the planar case. We also show that every branch point of a
quasiregular mapping with a small inner dilatation is a Lebesgue point of the
differential matrix of the mapping.
