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A remark on functions continuous on all lines. (English) Zbl 1474.26046
Summary: 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 [Real Anal. Exch. 41, No. 1, 19–54 (2016; Zbl 1388.26010)]. 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.
26B05 Continuity and differentiation questions
46B25 Classical Banach spaces in the general theory
Zbl 1388.26010
Full Text: DOI
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