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.
Let be a closed interval, a real-valued differentiable function. Then has the intermediate value property: If and are points in with , then for every between and , there exists an in such that .
Proof 1. The first proof is based on the extreme value theorem.
If equals or , then setting equal to or , respectively, gives the desired result. Now assume that is strictly between and , and in particular that . Let such that . If it is the case that we adjust our below proof, instead asserting that has its minimum on .
Since is continuous on the closed interval , the maximum value of on is attained at some point in , according to the extreme value theorem.
Because , we know cannot attain its maximum value at . (If it did, then for all , which implies .)
Likewise, because , we know cannot attain its maximum value at .
Therefore, must attain its maximum value at some point . Hence, by Fermat's theorem, , i.e. .
Define . For define and . And for define and .
Thus, for we have . Now, define with . is continuous in .
Furthermore, when and when ; therefore, from the Intermediate Value Theorem, if then, there exists such that . Let's fix .
From the Mean Value Theorem, there exists a point such that . Hence, .
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. 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.
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 function is a Darboux function even though it is not continuous at one point.
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. 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.
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- This article incorporates material from Darboux's theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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