Darboux function

In mathematics, a Darboux function, named for Michel Darboux, is a real-valued function f which has the "intermediate value property": on the interval between a and b, f assumes every real value between f(a) and f(b). Formally, for all real numbers a and b, and for every z such that f(a) < z < f(b), there exists some x with a < x < b such that f(x) = z.

By the intermediate value theorem, every continuous function is a Darboux function. Darboux's contribution was to show that there are discontinuous Darboux functions. Construction of a discontinuous Darboux function can proceed in at least two ways. One can use transfinite induction on Ω, or a construction involing Hamel bases.