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On sets of discontinuities of functions continuous on all lines

Luděk Zajíček (2022)

Commentationes Mathematicae Universitatis Carolinae

Answering a question asked by K. C. Ciesielski and T. Glatzer in 2013, we construct a C 1 -smooth function f on [ 0 , 1 ] and a closed set M ⊂ graph f nowhere dense in graph f such that there does not exist any linearly continuous function on ℝ 2 (i.e., function continuous on all lines) which is discontinuous at each point of M . We substantially use a recent full characterization of sets of discontinuity points of linearly continuous functions on ℝ n proved by T. Banakh and O. Maslyuchenko in 2020. As an easy consequence of our...

On smooth points of boundaries of open sets

S. Rolewicz (2009)

Studia Mathematica

The notions of smooth points of the boundary of an open set and α(·) intrinsically paraconvex sets are introduced. It is shown that for an α(·) intrinsically paraconvex open set the set of smooth points is a dense G δ -set of the boundary.

On Some Properties of Separately Increasing Functions from [0,1]ⁿ into a Banach Space

Artur Michalak (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

We say that a function f from [0,1] to a Banach space X is increasing with respect to E ⊂ X* if x* ∘ f is increasing for every x* ∈ E. A function f : [ 0 , 1 ] m → X is separately increasing if it is increasing in each variable separately. We show that if X is a Banach space that does not contain any isomorphic copy of c₀ or such that X* is separable, then for every separately increasing function f : [ 0 , 1 ] m → X with respect to any norming subset there exists a separately increasing function g : [ 0 , 1 ] m → ℝ such that the sets of points of discontinuity...

On weak sequential convergence in JB*-triple duals

Leslie J. Bunce, Antonio M. Peralta (2004)

Studia Mathematica

We study various Banach space properties of the dual space E* of a homogeneous Banach space (alias, a JB*-triple) E. For example, if all primitive M-ideals of E are maximal, we show that E* has the Alternative Dunford-Pettis property (respectively, the Kadec-Klee property) if and only if all biholomorphic automorphisms of the open unit ball of E are sequentially weakly continuous (respectively, weakly continuous). Those E for which E* has the weak* Kadec-Klee property are characterised by a compactness...

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