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.

We characterize those Baire one functions f for which the diagonal product x → (f(x), g(x)) has a connected graph whenever g is approximately continuous or is a derivative.

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).

The goal of this paper is to characterize the family of averages of comparable (Darboux) quasi-continuous functions.

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

We introduce and examine the notion of dense weak openness. In particular we show that multiplication in C(X) is densely weakly open whenever X is an interval in ℝ.

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