Linearly continuous maps discontinuous on the graphs of twice differentiable functions

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Krzysztof, Chris and Rodríguez-Vidanes, D.L. Linearly continuous maps discontinuous on the graphs of twice differentiable functions. (Unpublished)

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Abstract

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.


Item Type:Article
Uncontrolled Keywords:Differentiable nowhere monotone functions; Linearly continuous functions; Separate continuity
Subjects:Sciences > Mathematics
Sciences > Mathematics > Mathematical analysis
ID Code:73499
Deposited On:07 Jul 2022 11:07
Last Modified:07 Jul 2022 12:36

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