- Darboux function
In
mathematics , a Darboux function, named forGaston Darboux (1842-1917), is areal-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 allreal number s "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 , everycontinuous 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 usetransfinite 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 .
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 is a Darboux function that is not continuous.An example of a Darboux function that is nowhere continuous is the
Conway Base 13 function .
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