Restricted continuity and a theorem of Luzin

Krzysztof Chris Ciesielski ; Joseph Rosenblatt

Colloquium Mathematicae, Tome 135 (2014), p. 211-225 / Harvested from The Polish Digital Mathematics Library

Résumé

Let P(X,ℱ) denote the property: For every function f: X × ℝ → ℝ, if f(x,h(x)) is continuous for every h: X → ℝ from ℱ, then f is continuous. We investigate the assumptions of a theorem of Luzin, which states that P(ℝ,ℱ) holds for X = ℝ and ℱ being the class C(X) of all continuous functions from X to ℝ. The question for which topological spaces P(X,C(X)) holds was investigated by Dalbec. Here, we examine P(ℝⁿ,ℱ) for different families ℱ. In particular, we notice that P(ℝⁿ,"C¹") holds, where "C¹" is the family of all functions in C(ℝⁿ) having continuous directional derivatives allowing infinite values; and this result is the best possible, since P(ℝⁿ,D¹) is false, where D¹ is the family of all differentiable functions (no infinite derivatives allowed). We notice that if 𝓓 is the family of the graphs of functions from ℱ ⊆ C(X), then P(X,ℱ) is equivalent to the property P*(X,𝓓): For every f: X × ℝ → ℝ, if f↾ D is continuous for every D ∈ 𝓓, then f is continuous. Note that if 𝓓 is the family of all lines in ℝⁿ, then, for n ≥ 2, P*(ℝⁿ,𝓓) is false, since there are discontinuous linearly continuous functions on ℝⁿ. In this direction, we prove that there exists a Baire class 1 function h: ℝⁿ → ℝ such that P*(ℝⁿ,T(h)) holds, where T(H) stands for all possible translations of H ⊂ ℝⁿ × ℝ; and this result is the best possible, since P*(ℝⁿ,T(h)) is false for any h ∈ C(ℝⁿ). We also notice that P*(ℝⁿ,T(Z)) holds for any Borel Z ⊆ ℝⁿ × ℝ either of positive measure or of second category. Finally, we give an example of a perfect nowhere dense Z ⊆ ℝⁿ × ℝ of measure zero for which P*(ℝⁿ,T(Z)) holds.

Détail

Publié le : 2014-01-01

Zbl 1304.26008

EUDML-ID : urn:eudml:doc:284202