Darboux function

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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 $x \mapsto x^2\sin(1/x)$ is a Darboux function that is not continuous.