Linearly continuous maps discontinuous on the graphs of twice differentiable functions.

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]. [ABSTRACT FROM AUTHOR]