# 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.

## Proof

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

${\displaystyle \phi (t)=f(t)-yt.}$

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

By Darboux's theorem, the derivative of any differentiable function is a Darboux function. In particular, the derivative of the function ${\displaystyle x\mapsto x^{2}\sin(1/x)}$ 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

1. Olsen, Lars: A New Proof of Darboux's Theorem, Vol. 111, No. 8 (Oct., 2004) (pp. 713–715), The American Mathematical Monthly
2. {{#invoke:citation/CS1|citation |CitationClass=book }}
3. Bruckner, Andrew M: Differentiation of real functions, 2 ed, page 6, American Mathematical Society, 1994