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On the insertion of Darboux functions

Aleksander Maliszewski — 1998

Fundamenta Mathematicae

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.

Separating sets by Darboux-like functions

Aleksander Maliszewski — 2002

Fundamenta Mathematicae

We consider the following problem: Characterize the pairs ⟨A,B⟩ of subsets of ℝ which can be separated by a function from a given class, i.e., for which there exists a function f from that class such that f = 0 on A and f = 1 on B (the classical separation property) or f < 0 on A and f > 0 on B (a new separation property).

On theorems of Pu & Pu and Grande

Aleksander Maliszewski — 1996

Mathematica Bohemica

Given a finite family of cliquish functions, , we can find a Lebesgue function α such that f + α is Darboux and quasi-continuous for every f . This theorem is a generalization both of the theorem by H. W. Pu H. H. Pu and of the theorem by Z. Grande.

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