RT Journal Article
T1 Linearly continuous maps discontinuous on the graphs of twice differentiable functions
A1 Krzysztof, Chris
A1 Rodríguez-Vidanes, D.L.
AB A function g : R n → R is linearly continuous provided its restriction g   ` to every straight line ` ⊂ R n is continuous. It is known that the set D(g) of points of discontinuity of any linearly continuous g : R n → R is a countable union of isometric copies of (the graphs of) f   P, where f : R n−1 → R is Lipschitz and P ⊂ R n−1 is compact nowhere dense. On the other hand, for every twice continuously differentiable function f : R → R and every nowhere dense perfect P ⊂ R there is a linearly continuous g : R 2 → R with D(g) = f   P. The goal of this paper is to show that this last statement fails, if we do not assume that f 00 is continuous. More specifically, we show that this failure occurs for every continuously differentiable function f : R → R with nowhere monotone derivative, which includes twice differentiable functions f with such property. This generalizes a recent result of professor Ludek Zajicek and fully solves a problem from a 2013 paper of the first author and Timothy Glatzer.
LK https://hdl.handle.net/20.500.14352/65274
UL https://hdl.handle.net/20.500.14352/65274
LA eng
NO Ministerio de Ciencia e Innovación (MICINN)/FEDER
DS Docta Complutense
RD 8 abr 2025