Separate and joint properties of some analogues of pointwise discontinuity

Abstract

We study separate and joint properties of pointwise discontinuity, simple continuity and mildly continuity of functions of two variables. In particular it show that for a Baire space $X$, a Baire space $Y$ which has a countable pseudo-base and a metric space $Z$ a function $f:X\times Y\to Z$ is pointwise discontinuous if and only if $f$ satisfies $(\alpha, \beta)$-condition and condition (C) and $M=\{x\in X: \overline{C(f^x)}=Y\}$ is a residual subset of $X$. In addition, it was found of characterization of continuity for mappings of one and two variables