Linearly continuous maps discontinuous on the graphs of twice differentiable functions

Abstract:

A function $g\colon \mathbb {R}^n\to \mathbb {R}$ is linearly continuous provided its restriction $g\restriction \ell$ to every straight line $\ell \subset \mathbb {R}^n$ is continuous. It is known that the set $D(g)$ of points of discontinuity of any linearly continuous $g\colon \mathbb {R}^n\to \mathbb {R}$ is a countable union of isometric copies of (the graphs of) $f\restriction P$, where $f\colon \mathbb {R}^{n-1}\to \mathbb {R}$ is Lipschitz and $P\subset \mathbb {R}^{n-1}$ is compact nowhere dense. On the other hand, for every twice continuously differentiable function $f\colon \mathbb {R}\to \mathbb {R}$ and every nowhere dense perfect $P\subset \mathbb {R}$ there is a linearly continuous $g\colon \mathbb {R}^2\to \mathbb {R}$ with $D(g)=f\restriction P$. The goal of this paper is to show that this last statement fails, if we do not assume that $f''$ is continuous. More specifically, we show that this failure occurs for every continuously differentiable function $f\colon \mathbb {R}\to \mathbb {R}$ with nowhere monotone derivative, which includes twice differentiable functions $f$ with such property. This generalizes a recent result of professor Luděk Zajíček [On sets of discontinuities of functions continuous on all lines, arxiv.org/abs/2201.00772v1, 2022] and fully solves a problem from a 2013 paper of the first author and Timothy Glatzer [Real Anal. Exchange 38 (2012/13), pp. 377–389].