  We review results of {\v{S}}ver\'{a}k, and Goldstein-Haj{\l}asz-Pakzad on how
to show the continuity of functions in a critical Sobolev space with positive
Jacobian. In the final chapter we expand on the theory of $VMO$ functions,
showing a version of the change of variables theorem in this regularity, which
generalizes the first step in {\v{S}}ver\'{a}k's proof to a co-dimension 1
case.
