Discreteness and openness for mappings of finite distortion

Non-constant analytic functions map open sets to open sets and the preimage of a point is a discrete set of points. A natural generalization of the concept is that of a mapping of bounded distortion in the n-dimensional euclidean space: a continuous mapping whose first order distributional derivatives are n-integrable and so that the n-th power of the norm of the differential matrix Df(x) is almost everywhere controlled by a constant K multiple of the Jacobian determinant. By a result of Reshetnyak, the disceteness and openness holds for mappings of bounded distortion. We discuss (optimal) extensions of this result to the setting where K is allowed to depend also on the variable x. A mapping constructed by Ball appears to give the critical regularity of K.