×

zbMATH — the first resource for mathematics

A remark on functions continuous on all lines. (English) Zbl 1474.26046
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 [Real Anal. Exch. 41, No. 1, 19–54 (2016; Zbl 1388.26010)]. 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.
MSC:
26B05 Continuity and differentiation questions
46B25 Classical Banach spaces in the general theory
Citations:
Zbl 1388.26010
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ciesielski K. C.; Miller D., A continuous tale on continuous and separately continuous functions, Real Anal. Exchange 41 (2016), no. 1, 19-54
[2] Kershner R., The continuity of functions of many variables, Trans. Amer. Math. Soc. 53 (1943), 83-100
[3] Kuratowski K., Topology. Vol. I, Academic Press, New York, Państwowe Wydawnictwo Naukowe, Warszawa, 1966
[4] Lebesgue H., Sur les fonctions représentable analytiquement, J. Math. Pure Appl. (6) 1 (1905), 139-212 (French)
[5] Lukeš J.; Malý J.; Zajíček L., Fine Topology Methods in Real Analysis and Potential Theory, Lecture Notes in Mathematics, 1189, Springer, Berlin, 1986
[6] Massera J. L.; Schäffer J. J., Linear differential equations and functional analysis. I, Ann. of Math. (2) 67 (1958), 517-573
[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
[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
[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
[10] Zajíček L., On \(\sigma \)-porous sets in abstract spaces, Abstr. Appl. Anal. 2005 (2005), no. 5, 509-534
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.