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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.
Last updated: 10-25-2005 00:42:05
10-26-2009 08:16:03
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The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details