Real Analysis Exchange

Sets of Discontinuities of Linearly Continuous Functions

Abstract

The class of linearly continuous functions from $f \colon \mathbb{R}^n \to \mathbb{R}$, that is, having continuous restrictions $f \upharpoonright \ell$ to every straight line $\ell$, have been studied since the dawn of the twentieth century. In this paper we refine a description of the form that the sets $D(f)$ of points of discontinuities of such functions can have. It has been proved by Slobodnik that $D(f)$ must be a countable union of isometric copies of the graphs of Lipschitz functions $h\colon K\to\mathbb{R}$, where $K$ is a compact nowhere dense subset of $\mathbb{R}^{n-1}$. Since the class $\mathcal{D}^n$ of all sets $D(f)$, with $f \colon \mathbb{R}^n \to \mathbb{R}$ being linearly continuous, is evidently closed under countable unions as well as under isometric images, the structure of $\mathcal{D}^n$ will be fully discerned upon deciding precisely which graphs of the Lipschitz functions $h\colon K\to\mathbb{R}$, $K\subset\mathbb{R}^{n-1}$ being compact nowhere dense, belong to $\mathcal{D}^n$. Towards this goal, we prove that $\mathcal{D}^2$ contains the graph of any such $h\colon K\to\mathbb{R}$ whenever $h$ can be extended to a $C^2$ function $\bar h\colon\mathbb{R}\to\mathbb{R}$. Moreover, for every $n\gt 1$, $\mathcal{D}^n$ contains the graph of any $h\colon K\to\mathbb{R}$, where $K$ is closed nowhere dense in $\mathbb{R}^{n-1}$ and $h$ is a restriction of a convex function $\bar h\colon\mathbb{R}^{n-1}\to\mathbb{R}$. In addition, we provide an example, showing that the above mentioned result on $\mathcal{C}^2$ functions need not hold when $\bar h$ is just differentiable with bounded derivative (so Lipschitz).

Article information

Source
Real Anal. Exchange, Volume 38, Number 2 (2012), 377-390.

Dates
First available in Project Euclid: 27 June 2014