# A remark on functions continuous on all lines

Commentationes Mathematicae Universitatis Carolinae (2019)

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

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topZajíček, 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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