Darboux's theorem: Information from Answers.com

This article is about Darboux's theorem in real analysis. For Darboux's theorem in symplectic topology, see Darboux's theorem.

Darboux's theorem is a theorem in real analysis, named after Jean Gaston Darboux. It states that all functions which 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.

1 Darboux's theorem
2 Proof
3 Darboux Function
4 See also
5 External links

Darboux's theorem

Let f : [a,b] → R be a real-valued continuous function on [a,b], which is differentiable on (a,b), differentiable from the right at a, and differentiable from the left at b. Then $f\,'$ satisfies the intermediate value property: for every t between $f\,'_{+}(a)$ and $f\,'_{-}(b)$ , there is some x in [a,b] such that $f\,'(x) = t$ .

Proof

Without loss of generality we will assume $f\,'_{+}(a) > t > f\,'_{-}(b)$ . Let g(x) := f(x) - tx. Then $g'(x) = f\,'(x) - t$ exists by hypothesis, $g' + (a) > 0 > g' - (b)$ , and we wish to find a zero of $g'$ .

By hypothesis, g is a continuous function on [a,b], by the extreme value theorem it attains a maximum on [a,b]. This maximum cannot be at a, since $g' + (a) > 0$ so there should be a point $d > a$ with $g (d) > g (a)$ . Similarly, $g' - (b) < 0$ , so g cannot have a maximum at b. So the maximum is attained at some c in (a,b). But then $g'(c) = 0$ by Fermat's theorem (stationary points).

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. 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 $x \mapsto \sin(1/x)$ .

As a consequence of the mean value 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.

External links

This article incorporates material from Darboux's theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
This article incorporates material from Proof of Darboux's theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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Darboux's theorem

Contents

Darboux's theorem

Proof

Darboux Function

See also

External links