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 another function 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 theoremEdit

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]

ProofsEdit

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 functionEdit

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 topologist's sine curve 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. The Conway base 13 function is again an example.[4]

NotesEdit

  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 linksEdit