# Article

 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: http://hdl.handle.net/10338.dmlcz/147816 . Reference: [1] Ciesielski K. C., Miller D.: A continuous tale on continuous and separately continuous functions.Real Anal. Exchange 41 (2016), no. 1, 19–54. MR 3511935 Reference: [2] Kershner R.: The continuity of functions of many variables.Trans. Amer. Math. Soc. 53 (1943), 83–100. MR 0007522, 10.1090/S0002-9947-1943-0007522-5 Reference: [3] Kuratowski K.: Topology. Vol. I.Academic Press, New York, Państwowe Wydawnictwo Naukowe, Warszawa, 1966. Reference: [4] Lebesgue H.: Sur les fonctions représentable analytiquement.J. Math. Pure Appl. (6) 1 (1905), 139–212 (French). Reference: [5] Lukeš J., Malý J., Zajíček L.: Fine Topology Methods in Real Analysis and Potential Theory.Lecture Notes in Mathematics, 1189, Springer, Berlin, 1986. Zbl 0607.31001, MR 0861411, 10.1007/BFb0075905 Reference: [6] Massera J. L., Schäffer J. J.: Linear differential equations and functional analysis. I.Ann. of Math. (2) 67 (1958), 517–573. MR 0096985, 10.2307/1969871 Reference: [7] Shkarin S. A.: Points of discontinuity of Gateaux-differentiable mappings.Sibirsk. Mat. Zh. 33 (1992), no. 5, 176–185 (Russian); translation in Siberian Math. J. 33 (1992), no. 5, 905–913. MR 1197083 Reference: [8] Slobodnik S. G.: Expanding system of linearly closed sets.Mat. Zametki 19 (1976), 67–84 (Russian); translation in Math. Notes 19 (1976), 39–48. MR 0409742 Reference: [9] Zajíček L.: On the points of multivaluedness of metric projections in separable Banach spaces.Comment. Math. Univ. Carolin. 19 (1978), no. 3, 513–523. MR 0508958 Reference: [10] Zajíček L.: On $\sigma$-porous sets in abstract spaces.Abstr. Appl. Anal. 2005 (2005), no. 5, 509–534. Zbl 1098.28003, MR 2201041, 10.1155/AAA.2005.509 Reference: [11] Zajíček L.: Generic Fréchet differentiability on Asplund spaces via a.e. strict differentiability on many lines.J. Convex Anal. 19 (2012), no. 1, 23–48. MR 2934114 .

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