On symmetrical cliquishness and quasi-continuity of functions of two variables

摘要

We study the properties of joint cliquishness and quasi-continuity for functions of two variables. We introduce some properties (B) and (C) of functions of two variables such that (B) is an essential weakening of quasi-continuity and (C) is valid, in particular, for functions taking values in separable metrizable spaces. In particular, we prove the following theorem. Let X be a Baire space, Y a topological space which has a countable pseudo-base, Z a metric space and f : X ×Y → Z a function. Then a residual subset A of X such that f is symmetrically cliquish (quasi-continuous) with respect to x at each point of A × Y exists if and only if {x ∈ X : f~x is cliquish (quasi-continuous)} is residual in X and conditions (B) and (C) hold.