Restricted continuity and a theorem of Luzin
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...