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

Statement of the theorem

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

The original proof by Jean Gaston Darboux was published in 1875.

Proofs

Proof from the extreme value theorem

The first proof is based on the extreme value theorem.

If $y$ equals $f^{\prime }(a)$ or $f^{\prime }(b)$, then setting $x$ equal to $a$ or $b$, respectively, gives the desired result. Now assume that $y$ is strictly between $f^{\prime }(a)$ and $f^{\prime }(b)$, and in particular that $f^{\prime }(a)>y>f^{\prime }(b)$. Let $\phi :I\to ℝ$ such that $\phi (t)=f(t)-yt$. If it is the case that $f^{\prime }(a)<y<f^{\prime }(b)$ we adjust our below proof, instead asserting that $\phi$ has its minimum on $[a,b]$.

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

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

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

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

Proof from the mean and intermediate value theorems

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

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

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

Furthermore, $g(t)\to f^{\prime }(a)$ when $t\to a$ and $g(t)\to f^{\prime }(b)$ when $t\to b$; therefore, from the Intermediate Value Theorem, if $y\in (f^{\prime }(a),f^{\prime }(b))$ then, there exists $t_0\in (a,b)$ such that $g(t_0)=y$. Let's fix $t_0$.

From the Mean Value Theorem, there exists a point $x\in (\alpha (t_0),\beta (t_0))$ such that $f\text{'}(x)=g(t_0)$. Hence, $f\text{'}(x)=y$.

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 $y=f(c)$. 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:

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

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 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 $(x_i{)}_{i\in {\mathbb{Z}}_{+}}$ each 0 or 1, and $f(x)=\sum _{k=1}^{\mathrm{\infty }}\frac{(-1{)}^{x_k}}{k}$ if the series converges for that x and 0 if it does not.

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. 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.

Further restrictions on derivatives

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 $F_{\sigma }$ set. Conversely, every meagre $F_{\sigma }$ subset of the real line can occur as the discontinuity set of a derivative.

A finer restriction is on the sublevel sets of a derivative. For a real function $f$, its associated superlevel and sublevel sets are $\{x:f(x)>a\}$ and $\{x:f(x)<a\}$, where $a$ is real. Zahorski introduced classes $M_0,\dots ,M_5$ 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 $M_3$.
• Every bounded derivative has associated sets in $M_4$. Moreover, a set is an associated set of some bounded derivative if and only if it belongs to $M_4$.

Intuitively, if $f=F^{\prime }$ and $f(x_0)>a$, then the set on which $f>a$ cannot be arbitrarily sparse near $x_0$. If $f$ is continuous at $x_0$, this is trivial: $f>a$ throughout some neighbourhood of $x_0$, so the local density is $1$. The Zahorski conditions express weaker density requirements that remain valid even when the derivative is discontinuous.

More explicitly, a non-empty $F_{\sigma }$ set $E$ belongs to $M_3$ if, for every $x\in E$, any sequence of closed intervals $I_n$ not containing $x$, with $\mathrm{dist}(x,I_n)\to 0$ and $\lambda (I_n\cap E)=0$, satisfies

$\frac{\lambda (I_n)}{\mathrm{dist}(x,I_n)}\to 0,$

where $\lambda$ denotes Lebesgue measure. Thus, near a point of $E$, gaps in $E$ cannot have length comparable to their distance from the point. The class $M_4$ is stronger: $E$ belongs to $M_4$ if it can be written as a countable union of closed sets $E=\bigcup K_n$ such that, on each $K_n$, the set $E$ 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, $M_3$ rules out large nearby holes, while $M_4$ imposes a uniform positive lower-density condition.

External links

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