## Luděk Zajíček

*A remark on functions continuous on all lines*

Comment.Math.Univ.Carolin. 60,2 (2019) 211-218.**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.

**Keywords:** linear continuity; Baire class one; discontinuity set; Banach space

**DOI:** DOI 10.14712/1213-7243.2019.003

**AMS Subject Classification:** 26B05 46B99

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