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Title: A remark on functions continuous on all lines (English)
Author: Zajíček, Luděk
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 60
Issue: 2
Year: 2019
Pages: 211-218
Summary lang: English
Category: math
Summary: We prove that each linearly continuous function $f$ on $\mathbb R^n$ (i.e., each function continuous on all lines) belongs to the first Baire class, which answers a problem formulated by K.\,C. Ciesielski and D. Miller (2016). The same result holds also for $f$ on an arbitrary Banach space $X$, if $f$ has moreover the Baire property. We also prove (extending a known finite-dimensional result) that such $f$ on a separable $X$ is continuous at all points outside a first category set which is also null in any usual sense. (English)
Keyword: linear continuity
Keyword: Baire class one
Keyword: discontinuity set
Keyword: Banach space
MSC: 26B05
MSC: 46B99
idZBL: Zbl 07144890
idMR: MR3982469
DOI: 10.14712/1213-7243.2019.003
Date available: 2019-08-05T09:47:43Z
Last updated: 2021-07-05
Stable URL:
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