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We show that functions of two variables whose vertical sections are preponderantly continuous are strongly supmeasurable. [ABSTRACT FROM AUTHOR]

Making use of ideas of Marczewski and Sierpiński we propose a general approach to studies on connections between measurability, continuity and relative continuity of functions. Theorem 2.1 shows that a wellknown characterization of (s)-measurable Marczewski functions can be extended to the case of functions measurable with respect to a wide class of algebras involved with a topology. Theorem 2.2 generalizes the Denjoy-Stepanoff theorem and shows that the Denjoy-Stepanoff property stating the continuity of A-measurable functions at all points of a co-negligible set is quite common while an algebra A and an ideal J are the results of operations S and S0 on τ∖ I for a given topology τ . Also from the obtained results we conclude new theorems concerning the algebras associated with product ideals (Theorems 3.12 and 3.13). [ABSTRACT FROM AUTHOR]

A function f : ℝ → ℝ has the property 훭1 (훭2) if the restricted function f↾D(f) (f↾Dap(f)) is monotone. (D(f) [Dap(f)] denotes the set of all discontinuity points [the set of all approximate discontinuity points] of f.) In this article I investigate the uniform, pointwise and transfinite limits of sequences of functions with the property 훭i, i = 1, 2. [ABSTRACT FROM AUTHOR]

In this article we investigate the smallest (in the sense of inclusion) δ-field of subsets of R in which all functions of some families of functions from R to R satisfying some approximate quasicontinuity conditions introduced in [2] are measurable. [ABSTRACT FROM AUTHOR]