On embedding of the class .
On exponents of convergence of subsequences
On functions whose graph is of linear measure 0 on sets of measure 0
On Grande’s problem concerning functions
On jumping functions by connected sets
On Köpcke and Pompeiu functions
On level sets of Darboux functions
On Mauldin's classification of real functions
On pointwise limits of sequences of -continuous functions
On qualitative cluster sets
On rings of Darboux functions
On some new properties of the Cantor set
On some properties of the spaces of almost continuous functions.
On sums of Darboux functions
On symmetric derivatives and on properties of Zahorski
On the almost monotone convergence of sequences of continuous functions
A sequence (f n)n of functions f n: X → ℝ almost decreases (increases) to a function f: X → ℝ if it pointwise converges to f and for each point x ∈ X there is a positive integer n(x) such that f n+1(x) ≤ f n (x) (f n+1(x) ≥ f n(x)) for n ≥ n(x). In this article I investigate this convergence in some families of continuous functions.
On the closure of Baire classes under transfinite convergences
Let X be a Polish space and Y be a separable metric space. For a fixed ξ < ω₁, consider a family of Baire-ξ functions. Answering a question of Tomasz Natkaniec, we show that if for a function f: X → Y, the set is finite for every x ∈ X, then f itself is necessarily Baire-ξ. The proof is based on a characterization of sets which can be interesting in its own right.
On the continuity for connected functions
On the difference property of Borel measurable functions
If an atomlessly measurable cardinal exists, then the class of Lebesgue measurable functions, the class of Borel functions, and the Baire classes of all orders have the difference property. This gives a consistent positive answer to Laczkovich's Problem 2 [Acta Math. Acad. Sci. Hungar. 35 (1980)]. We also give a complete positive answer to Laczkovich's Problem 3 concerning Borel functions with Baire-α differences.
On the Dirichlet problem for functions of the first Baire class
Let be a simplicial function space on a metric compact space . Then the Choquet boundary of is an -set if and only if given any bounded Baire-one function on there is an -affine bounded Baire-one function on such that on . This theorem yields an answer to a problem of F. Jellett from [8] in the case of a metrizable set .