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 C^{1}([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]}
Proofs
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. .
Proof 2. The second proof is based on combining the mean value theorem and the intermediate value theorem.^{[1]}^{[2]}
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, .
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 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]}
Notes
- ^ ^{a} ^{b} Apostol, Tom M.: Mathematical Analysis: A Modern Approach to Advanced Calculus, 2nd edition, Addison-Wesley Longman, Inc. (1974), page 112.
- ^ ^{a} ^{b} Olsen, Lars: A New Proof of Darboux's Theorem, Vol. 111, No. 8 (Oct., 2004) (pp. 713–715), The American Mathematical Monthly
- ^ Rudin, Walter: Principles of Mathematical Analysis, 3rd edition, MacGraw-Hill, Inc. (1976), page 108
- ^ ^{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.
- ^ Bruckner, Andrew M: Differentiation of real functions, 2 ed, page 6, American Mathematical Society, 1994
External links
- This article incorporates material from Darboux's theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
- Hazewinkel, Michiel, ed. (2001) [1994], "Darboux theorem", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4