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Darboux's theorem (analysis)

Darboux's theorem (analysis)

In mathematics, Darboux's theorem is a theorem in real analysis, named after Jean Gaston Darboux. It states that every function that results from the differentiation of other functions has the intermediate value property: the image of an interval is also an interval.

When ƒ is continuously differentiable (ƒ in C1([a,b])), this is a consequence of the intermediate value theorem. But even when ƒ′ is not continuous, Darboux's theorem places a severe restriction on what it can be.

Darboux's theorem

Letbe aclosed interval,a real-valued differentiable function. Thenhas the intermediate value property: Ifandare points inwith, then for everybetweenand, there exists aninsuch that.[1][2][3]


Proof 1. The first proof is based on the extreme value theorem.

Ifequalsor, then settingequal toor, respectively, gives the desired result. Now assume thatis strictly betweenand, and in particular that. Letsuch that. If it is the case thatwe adjust our below proof, instead asserting thathas its minimum on.
Sinceis continuous on the closed interval, the maximum value ofonis attained at some point in, according to theextreme value theorem.
Because, we knowcannot attain its maximum value at. (If it did, thenfor all, which implies.)
Likewise, because, we knowcannot attain its maximum value at.
Therefore,must attain its maximum value at some point. Hence, byFermat's theorem,, i.e..

Proof 2. The second proof is based on combining the mean value theorem and the intermediate value theorem.[1][2]

Define. Fordefineand. And fordefineand.
Thus, forwe have. Now, definewith.is continuous in.
Furthermore,whenandwhen; therefore, from the Intermediate Value Theorem, ifthen, there existssuch that. Let's fix.
From the Mean Value Theorem, there exists a pointsuch that. Hence,.

Darboux function

A Darboux function is a real-valued function ƒ which has the "intermediate value property": for any two values a and b in the domain of ƒ, and any y between ƒ(a) and ƒ(b), there is some c between a and b with ƒ(c) = y.[4] By the intermediate value theorem, every continuous function on a real interval is a Darboux function. Darboux's contribution was to show that there are discontinuous Darboux functions.

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

By Darboux's theorem, the derivative of any differentiable function is a Darboux function. In particular, the derivative of the functionis a Darboux function even though it is not continuous at one point.

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

Darboux functions are a quite general class of functions. It turns out that any real-valued function ƒ on the real line can be written as the sum of two Darboux functions.[5] This implies in particular that the class of Darboux functions is not closed under addition.

A strongly Darboux function is one for which the image of every (non-empty) open interval is the whole real line. Such functions exist and are Darboux but nowhere continuous.[4]


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