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 .
On the family of functions with a closure of its graph of measure zero
On the Generalized Banach Indicatrix
On the insertion of Darboux, Baire-one functions
On the insertion of Darboux functions
The main goal of this paper is to characterize the family of all functions f which satisfy the following condition: whenever g is a Darboux function and f < g on ℝ there is a Darboux function h such that f < h < g on ℝ.
On the limits of sequences of Darboux a.e. quasi-continuous functions
On the pointwise limits of sequences of Świątkowski functions
The characterization of the pointwise limits of the sequences of Świątkowski functions is given. Modifications of Świątkowski property with respect to different topologies finer than the Euclidean topology are discussed.