On openness of density points under mappings (Q1396783)

The main result is the following theorem: Let \(A\) and \(B\) be Lebesgue measurable subsets of \(\mathbb{R}^n\), \(\tilde A\) and \(\tilde B\) be the sets of all density points of \(A\) and \(B\), respectively, \(x_0\in \tilde A\), \(y_0\in \tilde B\). Let \(f:\mathbb{R}^n\times\mathbb{R}^n \mapsto\mathbb{R}^n\) be of the class \(C^1\) on a neighbourhood of \((x_0,y_0)\) such that the Jacobi determinant of \(f(.,y_0)\) is nonzero in the point \(x_0\) and the Jacobi determinant of \(f(x_0,.)\) is nonzero in the point \(y_0\). Then \(f(x_0,y_0)\) is an interior point of \(f(\tilde A, \tilde B)\).