A-Darboux functions

We introduce and investigate the class of A-Darboux functions, namely, the class of functions f: ℝ→ℝ such that for all a, b ∈ ℝ with a < b and each y between f(a) and f(b), there is a point x 0 ∈ (a, b) ∩ A (where A is a nonempty fixed subset of ℝ) such that f(x 0) = y. Furthermore, we generalize the notion of the A-Darboux property for functions mapping a topological space into a topological space.