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On the insertion of Darboux functions

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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.
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Connectivity of diagonal products of Baire one functions

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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.
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Separating sets by Darboux-like functions

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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).
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On multiplication in spaces of continuous functions

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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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