Darboux's theorem (analysis)

In real analysis, Darboux's theorem states that the derivative of any real-valued function of a real variable has the intermediate value property, that is, that the image of an interval is also an interval.

When ${\displaystyle f}$ is continuously differentiable, this is a consequence of the intermediate value theorem. But even when ${\displaystyle f'}$ is not continuous, Darboux's theorem places a restriction on the behaviour of ${\displaystyle f'}$ over any closed interval.

Let ${\displaystyle I}$ be an open interval, and let ${\displaystyle f\colon I\to \mathbb {R} }$ be a real-valued differentiable function. Then ${\displaystyle f'}$ has the intermediate value property: If ${\displaystyle a}$ and ${\displaystyle b}$ are points in ${\displaystyle I}$ with ${\displaystyle a<b}$, then for every ${\displaystyle y}$ between ${\displaystyle f'(a)}$ and ${\displaystyle f'(b)}$, there exists an ${\displaystyle x}$ in ${\displaystyle [a,b]}$ such that ${\displaystyle f'(x)=y}$.^[1][2][3]

The original proof by Jean Gaston Darboux has been published in 1875.^[4]

A Darboux function is a real-valued function${\displaystyle f}$ which has the "intermediate value property": for any two values ${\displaystyle a}$ and ${\displaystyle b}$ in the domain of ${\displaystyle f}$, and any ${\displaystyle y}$ between ${\displaystyle f(a)}$ and ${\displaystyle f(b)}$, there is some ${\displaystyle c}$ between ${\displaystyle a}$ and ${\displaystyle b}$ with ${\displaystyle y=f(c)}$.^[5] By the intermediate value theorem, every continuous function on a realinterval 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 topologist's sine curve function:

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

By Darboux's theorem, the derivative of any differentiable function is a Darboux function. In particular, the derivative of the function ${\displaystyle x\mapsto x^{2}\sin(1/x)}$ is a Darboux function even though it is not continuous at one point.

An example of a Darboux function that is nowhere continuous is Conway's base 13 function. Another is Bergfeldt's function where a real number x is written in expanded in binary with digits ${\displaystyle (x_{i})_{i\in \mathbb {Z} _{+}}}$ each 0 or 1, and ${\displaystyle f(x)=\sum \limits _{k=1}^{\infty }{\frac {(-1)^{x_{k}}}{k}}}$ if the series converges for that x and 0 if it does not.^[6]

Darboux functions are a quite general class of functions. It turns out that any real-valued function ƒ on the real line can be written as the sum of two Darboux functions.^[7] 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.^[5]

Darboux's theorem gives a necessary condition for a function to be a derivative, but it is not sufficient. Every derivative of a real function is also of Baire class one, and the set of points at which a derivative is discontinuous is a meagre${\displaystyle F_{\sigma }}$ set. Conversely, every meagre ${\displaystyle F_{\sigma }}$ subset of the real line can occur as the discontinuity set of a derivative.^[8]

A finer restriction is on the sublevel sets of a derivative. For a real function ${\displaystyle f}$, its associated superlevel and sublevel sets are ${\displaystyle \{x:f(x)>a\}}$ and ${\displaystyle \{x:f(x)<a\}}$, where ${\displaystyle a}$ is real. Zahorski introduced classes ${\displaystyle M_{0},\ldots ,M_{5}}$ of sets describing how large such associated sets must be near their own points. In this terminology, one has the following theorems:

• Every finite derivative has associated sets in ${\displaystyle M_{3}}$.
• Every bounded derivative has associated sets in ${\displaystyle M_{4}}$. Moreover, a set is an associated set of some bounded derivative if and only if it belongs to ${\displaystyle M_{4}}$.^[9]

Intuitively, if ${\displaystyle f=F'}$ and ${\displaystyle f(x_{0})>a}$, then the set on which ${\displaystyle f>a}$ cannot be arbitrarily sparse near ${\displaystyle x_{0}}$. If ${\displaystyle f}$ is continuous at ${\displaystyle x_{0}}$, this is trivial: ${\displaystyle f>a}$ throughout some neighbourhood of ${\displaystyle x_{0}}$, so the local density is ${\displaystyle 1}$. The Zahorski conditions express weaker density requirements that remain valid even when the derivative is discontinuous.

More explicitly, a non-empty ${\displaystyle F_{\sigma }}$ set ${\displaystyle E}$ belongs to ${\displaystyle M_{3}}$ if, for every ${\displaystyle x\in E}$, any sequence of closed intervals ${\displaystyle I_{n}}$ not containing ${\displaystyle x}$, with ${\displaystyle \operatorname {dist} (x,I_{n})\to 0}$ and ${\displaystyle \lambda (I_{n}\cap E)=0}$, satisfies

${\displaystyle {\frac {\lambda (I_{n})}{\operatorname {dist} (x,I_{n})}}\to 0,}$

where ${\displaystyle \lambda }$ denotes Lebesgue measure. Thus, near a point of ${\displaystyle E}$, gaps in ${\displaystyle E}$ cannot have length comparable to their distance from the point. The class ${\displaystyle M_{4}}$ is stronger: ${\displaystyle E}$ belongs to ${\displaystyle M_{4}}$ if it can be written as a countable union of closed sets ${\displaystyle E=\bigcup K_{n}}$ such that, on each ${\displaystyle K_{n}}$, the set ${\displaystyle E}$ occupies a uniformly positive proportion of every sufficiently small one-sided interval whose length is comparable with its distance from the point. In this sense, ${\displaystyle M_{3}}$ rules out large nearby holes, while ${\displaystyle M_{4}}$ imposes a uniform positive lower-density condition.

• This article incorporates material from Darboux's theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
• "Darboux theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]