Abstrakt
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.