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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Linearly continuous maps discontinuous on the graphs of twice differentiable functions
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by Krzysztof Chris Ciesielski and Daniel L. Rodríguez-Vidanes
Proc. Amer. Math. Soc. 151 (2023), 1979-1986
DOI: https://doi.org/10.1090/proc/16235
Published electronically: February 28, 2023

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].
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Bibliographic Information
  • Krzysztof Chris Ciesielski
  • Affiliation: Department of Mathematics, West Virginia University, Morgantown, West Virginia 26506-6310
  • MR Author ID: 49415
  • ORCID: 0000-0003-4282-5855
  • Email: KCiesiel@mix.wvu.edu
  • Daniel L. Rodríguez-Vidanes
  • Affiliation: Instituto de Matemática Interdisciplinar, Departamento de Análisis Matemático y Matemática Aplicada, Facultad de Ciencias Matemáticas, Plaza de Ciencias 3, Universidad Complutense de Madrid, Madrid 28040, Spain
  • MR Author ID: 1300751
  • ORCID: 0000-0002-1016-096X
  • Email: dl.rodriguez.vidanes@ucm.es
  • Received by editor(s): May 12, 2022
  • Received by editor(s) in revised form: July 15, 2022
  • Published electronically: February 28, 2023
  • Additional Notes: The second author was supported by Grant PGC2018-097286-B-I00 and by the Spanish Ministry of Science, Innovation and Universities and the European Social Fund through a “Contrato Predoctoral para la Formación de Doctores, 2019” (PRE2019-089135). The paper was partially done at West Virginia University (WVU) during a stay of the second author, who thanks the support provided by the Department of Mathematics at WVU
  • Communicated by: Nageswari Shanmugalingam
  • © Copyright 2023 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 151 (2023), 1979-1986
  • MSC (2020): Primary 26B05; Secondary 26A24
  • DOI: https://doi.org/10.1090/proc/16235
  • MathSciNet review: 4556193