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

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