Whitney sets and sets of constancy. On a problem of Whitney
A connected subset H of \(R^ n\) is called a Whitney set (W-set) iff there exists a nonconstant function \(f:H\to R\) such that \(\lim_{x\to a}| \frac{f(x)-f(a)}{x-a}| =0\) holds for every \(a\in H.\) Every connected not W-subset of \(R^ n\) is said to be a set of constancy (C- set). There are presented the following theorems: 1. There exists an one-to-one continuous mapping \(f:<0,1>\to R^ 2\) such that \(\lim_{x\to a}| \frac{f(x)-f(a)}{x-a}| =\infty\) holds uniformly for every \(a\in <0,1>.\) 2. Let \(f:<0,1>\to R^ n\) be a parametrization of a simple arc \(\gamma \subset R^ n.\) If the Lebesgue measure of the set \(\{x\in <0,1>:\lim_{y\to x}| \frac{f(y)-f(x)}{y- x}| =\infty \}\) is positive then \(\gamma\) is a W-set. 3. If \(f:<0,1>\to R^ n\) is continuous and if the set \(\{f(x):x\in <0,1>,\lim_{y\to x+}| \frac{f(y)-f(x)}{y-x}| =\infty \}\) has a \(\sigma\)-finite linear measure then \(f(<0,1>)\) is a C-set. From Theorem 1 it follows easily that the set \(f(<0,1>)\) is a W-set and Theorem 3 is a generalization of Choquet's theorem that the graph of any real continuous function defined on \(<0,1>\) is a C-set.
scientific article; zbMATH DE number 3953131