Let be a domain. Suppose that f ∈ W1,1loc(Ω,R2) is a homeomorphism such that Df(x) vanishes almost everywhere in the zero set of J f . We show that f-1 ∈ W1,1loc(f(Ω),R2) and that Df−1(y) vanishes almost everywhere in the zero set of Sharp conditions to quarantee that f−1 ∈ W1,q(f(Ω),R2) for some 1<q≤2 are also given.