Local and global inversion theorems without assuming continuous differentiability. (English) Zbl 0721.30015

Cristea, Mihai

Bull. Math. Soc. Sci. Math. Répub. Soc. Roum., Nouv. Sér. 33(81), No. 3, 233-238 (1989).

The main result of this article is the following Theorem 1: Let \(n\geq 3\), \(D\subset {\mathbb{R}}^ n\) be open, \(K\subset D\) such that \(K\neq D\). Let \(K=\cap^{\infty}_{p=1}K_ p\), where \(K_ p\) is a closed set for every p, f: \(D\to {\mathbb{R}}^ n\) continuous and light such that \(m_{n- 2}(f(K_ p))=0\) for every \(p\in {\mathbb{N}}\) and the function f is differentiable on \(D\setminus K\) and \(J_ f(x)\neq 0\) for every \(x\in D\setminus K\). Then f is a local homeomorphism on D. As corollaries of this result the author obtains some theorems about global homeomorphisms. For example, Theorem 4. Let \(n\geq 3\), E, F be open and pathwise connected and \(K\subset E\) such that \(K\neq E\). \(K=\cap^{\infty}_{p=1}K_ p\), where \(K_ p\) is a closed set for every p, f: \(E\to F\) is continuous, closed and light such that \(m_{n-2}(f(K_ p))=0\) for every \(p\in {\mathbb{N}}\) and f is differentiable on \(E\setminus K\) and \(J_ f(x)\neq 0\) for every \(x\in E\setminus K\). Then f: \(E\to F\) is a global homeomorphism.

MSC:

30C65

Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations