Darboux's theorem (analysis)
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 f is continuously differentiable (f in C^{1}([a,b])), this is a consequence of the intermediate value theorem. But even when f′ is not continuous, Darboux's theorem places a severe restriction on what it can be.
Darboux's theorem
Let be an open 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]}
Proof
If equals or , then setting equal to or , respectively, works. Therefore, without loss of generality, we may assume that is strictly between and , and in particular that . Define a new function by
Since is continuous on the closed interval , its maximum value on that interval is attained, according to the extreme value theorem, at a point in that interval, i.e. at some . Because , we see cannot attain its maximum value on at . Because , we see cannot attain its maximum value on at . Therefore, . Hence, by Fermat's theorem, , i.e. .
Another proof can be given by combining the mean value theorem and the intermediate value theorem.^{[2]}^{[1]}
Darboux function
A Darboux function is a real-valued function f which has the "intermediate value property": for any two values a and b in the domain of f, and any y between f(a) and f(b), there is some c between a and b with f(c) = y.^{[3]} 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 f on the real line can be written as the sum of two Darboux functions.^{[4]} 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.^{[3]}
Notes
- 1 2 Olsen, Lars: A New Proof of Darboux's Theorem, Vol. 111, No. 8 (Oct., 2004) (pp. 713–715), The American Mathematical Monthly
- ↑ Apostol, Tom M.: Mathematical Analysis: A Modern Approach to Advanced Calculus, 2nd edition, Addison-Wesley Longman, Inc. (1974), page 112.
- 1 2 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), "Darboux theorem", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4