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 $I$ 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 $a<b$, then for every $y$ between $f'(a)$ and $f'(b)$, there exists an $x$ in $[a,b]$ such that $f'(x)=y$.^[1]

Proof

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

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 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

$x \mapsto \begin{cases}\sin(1/x)& \text{for } x\ne 0\\ 0 &\text{for }x=0\end{cases}$.

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 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.^[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

1. Olsen, Lars: A New Proof of Darboux's Theorem, Vol. 111, No. 8 (Oct., 2004) (pp. 713–715), The American Mathematical Monthly
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.
3. 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