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 ${\displaystyle f'(x)=y}$.¹²³

The original proof by Jean Gaston Darboux has been published in 1875.⁴

The first proof is based on the extreme value theorem.

If ${\displaystyle y}$ equals ${\displaystyle f'(a)}$ or ${\displaystyle f'(b)}$, then setting ${\displaystyle x}$ equal to ${\displaystyle a}$ or ${\displaystyle b}$, respectively, gives the desired result. Now assume that ${\displaystyle y}$ is strictly between ${\displaystyle f'(a)}$ and ${\displaystyle f'(b)}$, and in particular that ${\displaystyle f'(a)>y>f'(b)}$. Let ${\displaystyle \varphi \colon I\to \mathbb {R} }$ such that ${\displaystyle \varphi (t)=f(t)-yt}$. If it is the case that ${\displaystyle f'(a)<y<f'(b)}$ we adjust our below proof, instead asserting that ${\displaystyle \varphi }$ has its minimum on ${\displaystyle [a,b]}$.

Since ${\displaystyle \varphi }$ is continuous on the closed interval ${\displaystyle [a,b]}$, the maximum value of ${\displaystyle \varphi }$ on ${\displaystyle [a,b]}$ is attained at some point in ${\displaystyle [a,b]}$, according to the extreme value theorem.

Because ${\displaystyle \varphi '(a)=f'(a)-y>0}$, we know ${\displaystyle \varphi }$ cannot attain its maximum value at ${\displaystyle a}$. (If it did, then ${\displaystyle (\varphi (t)-\varphi (a))/(t-a)\leq 0}$ for all ${\displaystyle t\in (a,b]}$, which implies ${\displaystyle \varphi '(a)\leq 0}$.)

Likewise, because ${\displaystyle \varphi '(b)=f'(b)-y<0}$, we know ${\displaystyle \varphi }$ cannot attain its maximum value at ${\displaystyle b}$.

Therefore, ${\displaystyle \varphi }$ must attain its maximum value at some point ${\displaystyle x\in (a,b)}$. Hence, by Fermat's theorem, ${\displaystyle \varphi '(x)=0}$, i.e. ${\displaystyle f'(x)=y}$.

The second proof is based on combining the mean value theorem and the intermediate value theorem.¹²

Define ${\displaystyle c={\frac {1}{2}}(a+b)}$. For ${\displaystyle a\leq t\leq c,}$ define ${\displaystyle \alpha (t)=a}$ and ${\displaystyle \beta (t)=2t-a}$. And for ${\displaystyle c\leq t\leq b,}$ define ${\displaystyle \alpha (t)=2t-b}$ and ${\displaystyle \beta (t)=b}$.

Thus, for ${\displaystyle t\in (a,b)}$ we have ${\displaystyle a\leq \alpha (t)<\beta (t)\leq b}$. Now, define ${\displaystyle g(t)={\frac {(f\circ \beta )(t)-(f\circ \alpha )(t)}{\beta (t)-\alpha (t)}}}$ with ${\displaystyle a<t<b}$. ${\displaystyle \,g}$ is continuous in ${\displaystyle (a,b)}$.

Furthermore, ${\displaystyle g(t)\rightarrow {f}'(a)}$ when ${\displaystyle t\rightarrow a}$ and ${\displaystyle g(t)\rightarrow {f}'(b)}$ when ${\displaystyle t\rightarrow b}$; therefore, from the Intermediate Value Theorem, if ${\displaystyle y\in ({f}'(a),{f}'(b))}$ then, there exists ${\displaystyle t_{0}\in (a,b)}$ such that ${\displaystyle g(t_{0})=y}$. Let's fix ${\displaystyle t_{0}}$.

From the Mean Value Theorem, there exists a point ${\displaystyle x\in (\alpha (t_{0}),\beta (t_{0}))}$ such that ${\displaystyle {f}'(x)=g(t_{0})}$. Hence, ${\displaystyle {f}'(x)=y}$.

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 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 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.