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 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

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.

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.^[1] This implies in particular that the class of Darboux functions is not closed under addition.

Notes

1. ^ 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.
• This article incorporates material from Proof of Darboux's theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.