## Luděk ZajíčekA 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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