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 f is continuously differentiable (f in C1([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[edit]

Let I be an open interval, f\colon I\to \R a real-valued differentiable function. Then f' has the intermediate value property: If a and b are points in I with a\leq b, then for every y between f'(a) and f'(b), there exists an x in [a,b] such that f'(x)=y.[1]

Proof[edit]

If y equals f'(a) or f'(b), then setting x equal to a or b, respectively, works. Therefore, without loss of generality, we may assume that y is strictly between f'(a) and f'(b), and in particular that f'(a)>y>f'(b). Define a new function \phi\colon I\to \R by

\phi(t)=f(t)-yt.

Since \phi is continuous on the closed interval [a,b], its maximum value on that interval is attained, according to the extreme value theorem, at a point x in that interval, i.e. at some x\in[a,b]. Because \phi'(a)=f'(a)-y>y-y=0 and \phi'(b)=f'(b)-y<y-y=0, Fermat's theorem implies that neither a nor b can be a point, such as x, at which \phi attains a local maximum. Therefore, x\in(a,b). Hence, again by Fermat's theorem, \phi'(x)=0, i.e. f'(x)=y.[1]

Another proof based solely on the mean value theorem and the intermediate value theorem is due to Lars Olsen.[1]

Darboux function[edit]

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.[2] By the intermediate value theorem, every continuous function 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 x \mapsto \sin(1/x).

By Darboux's theorem, the derivative of any differentiable function is a Darboux function. In particular, the derivative of the function x \mapsto x^2\sin(1/x) is a Darboux function that is not continuous.

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.[3] 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.[2]

Notes[edit]

  1. ^ a b c Olsen, Lars: A New Proof of Darboux's Theorem, Vol. 111, No. 8 (Oct., 2004) (pp. 713-715), The American Mathematical Monthly
  2. ^ 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. 
  3. ^ Bruckner, Andrew M: Differentiation of real functions, 2 ed, page 6, American Mathematical Society, 1994

External links[edit]