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
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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].References
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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