A remark on functions continuous on all lines

Luděk Zajíček

Commentationes Mathematicae Universitatis Carolinae (2019)

  • Volume: 60, Issue: 2, page 211-218
  • ISSN: 0010-2628

Abstract

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We prove that each linearly continuous function f on 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.

How to cite

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Zajíček, Luděk. "A remark on functions continuous on all lines." Commentationes Mathematicae Universitatis Carolinae 60.2 (2019): 211-218. <http://eudml.org/doc/294315>.

@article{Zajíček2019,
abstract = {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.},
author = {Zajíček, Luděk},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {linear continuity; Baire class one; discontinuity set; Banach space},
language = {eng},
number = {2},
pages = {211-218},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A remark on functions continuous on all lines},
url = {http://eudml.org/doc/294315},
volume = {60},
year = {2019},
}

TY - JOUR
AU - Zajíček, Luděk
TI - A remark on functions continuous on all lines
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2019
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 60
IS - 2
SP - 211
EP - 218
AB - 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.
LA - eng
KW - linear continuity; Baire class one; discontinuity set; Banach space
UR - http://eudml.org/doc/294315
ER -

References

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  1. Ciesielski K. C., Miller D., A continuous tale on continuous and separately continuous functions, Real Anal. Exchange 41 (2016), no. 1, 19–54. MR3511935
  2. Kershner R., 10.1090/S0002-9947-1943-0007522-5, Trans. Amer. Math. Soc. 53 (1943), 83–100. MR0007522DOI10.1090/S0002-9947-1943-0007522-5
  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., 10.1007/BFb0075905, Lecture Notes in Mathematics, 1189, Springer, Berlin, 1986. Zbl0607.31001MR0861411DOI10.1007/BFb0075905
  6. Massera J. L., Schäffer J. J., 10.2307/1969871, Ann. of Math. (2) 67 (1958), 517–573. MR0096985DOI10.2307/1969871
  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. MR1197083
  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. MR0409742
  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. MR0508958
  10. Zajíček L., 10.1155/AAA.2005.509, Abstr. Appl. Anal. 2005 (2005), no. 5, 509–534. Zbl1098.28003MR2201041DOI10.1155/AAA.2005.509
  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. MR2934114

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