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).