Invertibility and a topological property of Sobolev maps

Let O be a bounded domain in Rn, let d : O¯ ¿ O¯ be a homeomorphism, and consider a function u : O¯ ¿ Rn that agrees with d on ¿O. If u is continuous and injective then u(O) = d(O). Motivated by problems in nonlinear elasticity the relationship between u(O) and d(O) is analyzed when the continuity and invertibility assumptions are weakened. Specifically maps that are continuous on almost every line and maps that lie in the Sobolev space W1,p with n - 1 < p < n are considered.