Darboux's theorem (analysis)

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Darboux's theorem is a theorem in real analysis, named after Jean Gaston Darboux. It states that all functions that result from the differentiation of other functions have 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

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 .[1][2][3]

Proof

If equals or , then setting equal to or , respectively, works. 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 . 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. .

Another proof can be given by combining the mean value theorem and the intermediate value theorem.[1][2]

In fact, let's take . For define and . And for define and .

Thus, for we have . Now, define with . is continuous in .

Furthermore, when and when , and, 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, .

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 function is a Darboux function that 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]

Notes

  1. ^ a b Apostol, Tom M.: Mathematical Analysis: A Modern Approach to Advanced Calculus, 2nd edition, Addison-Wesley Longman, Inc. (1974), page 112.
  2. ^ a b Olsen, Lars: A New Proof of Darboux's Theorem, Vol. 111, No. 8 (Oct., 2004) (pp. 713–715), The American Mathematical Monthly
  3. ^ Rudin, Walter: Principles of Mathematical Analysis, 3rd edition, MacGraw-Hill, Inc. (1976), page 108
  4. ^ a b Ciesielski, Krzysztof (1997). Set theory for the working mathematician. London Mathematical Society Student Texts. 39. Cambridge: Cambridge University Press. pp. 106–111. ISBN 0-521-59441-3. Zbl 0938.03067. 
  5. ^ Bruckner, Andrew M: Differentiation of real functions, 2 ed, page 6, American Mathematical Society, 1994

External links

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