Sectionally continuous injections of Euclidean spaces (Q908538)

A function f: \({\mathbb{R}}^ n\to Y\), \(n\geq 2\), is sectionally continuous if for each (n-1)-dimensional hyperplane H in \({\mathbb{R}}^ n\), \(f| H\) is continuous. The authors show that if \(f:{\mathbb{R}}^ n\to S^ n\) is a sectionally continuous injection and x is in \({\mathbb{R}}^ n\), then f is continuous at x if and only if f(x) is not the limit point of any component of \(S^ n\setminus f({\mathbb{R}}^ n)\). As a corollary, f is an embedding if and only if \(f({\mathbb{R}}^ n)\) is open. The authors also produce an example of a bijection f: \({\mathbb{R}}^ 3\to {\mathbb{R}}^ 3\) such that for each line L in \({\mathbb{R}}^ 3\), both \(f| L\) and \(f^{- 1}| L\) are piecewise linear homeomorphisms, but such that f is not continuous. For \(n=2\), the authors describe all possible images for sectionally continuous injections with only countably many discontinuities.