On sets of discontinuities of functions continuous on all lines

TL;DR

This paper constructs a smooth function with a nowhere dense set of discontinuities on its graph, demonstrating that certain necessary conditions for such discontinuity sets are not sufficient, and extends results to Banach spaces.

Contribution

It provides a counterexample to a 1976 necessary condition for discontinuity sets of linearly continuous functions, using recent characterizations, and extends the analysis to Banach spaces.

Findings

Constructed a $C^1$-smooth function with a nowhere dense discontinuity set.

Showed the necessary condition by Slobodnik is not sufficient.

Extended the results to separable Banach spaces.

Abstract

Answering a question asked by K.C. Ciesielski and T. Glatzer in 2013, we construct a $C^1$-smooth function $f$ on $[0,1]$ and a set $M \subset \operatorname{graph} f$ nowhere dense in $\operatorname{graph} f$ such that there does not exist any linearly continuous function on $\mathbb R^2$ (i.e. function continuous on all lines) which is discontinuous at each point of $M$. We substantially use a recent full characterization of sets of discontinuity points of linearly continuous functions on $\mathbb R^n$ proved by T. Banakh and O. Maslyuchenko in 2020. As an easy consequence of our result, we prove that the necessary condition for such sets of discontinuities proved by S.G. Slobodnik in 1976 is not sufficient. We also prove an analogon of this Slobodnik's result in separable Banach spaces.