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

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→Zf: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×YA×Y exists if and only if {x∈X:fx is cliquish (quasi-continuous)}{x∈X:fx is cliquish (quasi-continuous)} is residual in X and conditions (B) and (C) hold.