Darboux function

In mathematics, a Darboux function, named for Gaston Darboux (1842-1917), 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 involving Hamel bases.

Every discontinuity of a Darboux function is essential, that is, at any point of discontinuity, at least one of the left hand and right hand limits does not exist.

An example of a Darboux function that is discontinuous at one point, is the function $x\; mapsto\; sin(1/x)$.

As a consequence of the mean value theorem, the derivative of any differentiable function is a Darboux function. In particular, the derivative of the function $x\; mapsto\; x^2sin(1/x)$ is a Darboux function that is not continuous.

An example of a Darboux function that is nowhere continuous is the Conway Base 13 function.