The discontinuity point sets of quasi-continuous functions

Maslyuchenko, O. V.

doi:10.1017/s0004972700039307

Cited by 5 publications

(4 citation statements)

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“…Theorem 9.9 is not true if X is normal is replaced with X is T 1 completely regular, as the Niemytzki plane shows ( [9]). Other characterization (for spaces not Baire only) we can find in [48].…”

Section: Quasicontinuity and Cliquishnessmentioning

confidence: 96%

“…Theorem 9.10 ( [48]). Let for a Fréchet-Urysohn space X at least one of the following conditions holds: (i) X is a hereditarily separable perfectly normal; (ii) X is hereditarily quasi-separable perfectly normal; (iii) X is a regular space with a countable net; (iv) X is a paracompact with a σ -locally finite net; (v) X is metrizable.…”

Section: Quasicontinuity and Cliquishnessmentioning

confidence: 99%

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Points of quasicontinuity and of similar generalizations of continuity

Borsík¹

2013

Traditional and Present-Day Topics in Real Analysis. Dedicated to Professor Jan Stanisław Lipiński

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Section: Quasicontinuity and Cliquishnessmentioning

confidence: 96%

Section: Quasicontinuity and Cliquishnessmentioning

confidence: 99%

Points of quasicontinuity and of similar generalizations of continuity

Borsík¹

2013

Traditional and Present-Day Topics in Real Analysis. Dedicated to Professor Jan Stanisław Lipiński

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“…There are many results describing the sets of discontinuity points of functions from various function classes (see for example [6,14,16,19,21,22,24]) In Section 6 we shall prove the following description of the sets of discontinuity points of linearly continuous functions.…”

Section: Introductionmentioning

confidence: 99%

Linearly continuous functions and $F_σ$-measurability

Banakh,

Maslyuchenko

2019

Preprint

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The linear continuity of a function defined on a vector space means that its restriction to every affine line is continuous. For functions defined on R m this notion is near to the separate continuity for which it is required only the continuity on the straight lines which are parallel to coordinate axes. The classical Lebesgue theorem states that every separately continuous function f : R m → R is of the (m − 1)-th Baire class. In this paper we prove that every linearly continuous function f : R m → R is of the first Baire class. Moreover, we obtain the following result. If X is a Baire cosmic topological vector space, Y is a Tychonoff topological space and f : X → Y is a Borel-measurable (even BP-measurable) linearly continuous function, then f is Fσ-measurable. Using this theorem we characterize the discontinuity point set of an arbitrary linearly continuous function on R m . In the final part of the article we prove that any Fσ-measurable function f : ∂U → R defined on the boundary of a strictly convex open set U ⊂ R m can be extended to a linearly continuous function f : X → R. This fact shows that in the "descriptive sense" the linear continuity is not better than the Fσ-measurability.

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“…On the other hand, if X = R 2 [19] or if X is a Baire pseudometrizable space space without isolated points (or X is a Baire resolvable perfectly normal locally connected space) [5] or X is a hereditarily separable perfectly normal Fréchet-Urysohn space [50], then for each F σ -set A of first category there is a quasicontinuous function f : X → R such that A is the set of all discontinuity points of this function. Points of quasicontinuity were characterized in [45].…”

Section: Introductionmentioning

confidence: 99%