Restricted continuity and a theorem of Luzin

Abstract

Let P (X,F) denotes the property: For every function f : X × R → R, if f(x, h(x)) is continuous for every function h : X → R from F , then f is continuous. In this paper we investigate the assumptions of a theorem of Luzin [13], 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 in [5]. Here, we examine P (R,F) for different families F . In particular, we notice that: (1) P (R, “C”) holds, where “C” is the family of all functions in C(R) having continuous directional derivatives, allowing infinite values; (2) this result is the best possible, since P (R, D) is false, where D 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 function 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 in R, then, for n ≥ 2, P ∗(Rn,D) is false, since there are discontinuous linearly continuous functions on R. In this direction, we prove that: (1) there exists a Baire class 1 function h : R → R such that P ∗(Rn, T (h)) holds, where T (H) stands for all possible translations of H ⊂ R × R; (2) this result is the best possible, since P ∗(Rn, T (h)) is false for any h ∈ C(R). We also notice that P ∗(Rn, T (Z)) holds for any Borel Z ⊆ R × R either of positive measure or of second category. Finally, we give an example of a perfect nowhere dense Z ⊆ R × R of measure zero for which P ∗(Rn, T (Z)) holds.