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