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linear continuity; Baire class one; discontinuity set; Banach space
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.
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