Darboux's theorem (analysis)

Proofs

Proof from the extreme value theorem

The first proof is based on the extreme value theorem.

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

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

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

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

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

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

Thus, for ${\displaystyle t\in (a,b)}$ we have ${\displaystyle a\le \alpha (t)<\beta (t)\le b}$. 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, ${\displaystyle g(t)\to f^{\prime }(a)}$ when ${\displaystyle t\to a}$ and ${\displaystyle g(t)\to f^{\prime }(b)}$ when ${\displaystyle t\to b}$; therefore, from the Intermediate Value Theorem, if ${\displaystyle y\in (f^{\prime }(a),f^{\prime }(b))}$ 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 ${\displaystyle f^{\prime }(x)=g(t_0)}$. Hence, ${\displaystyle f^{\prime }(x)=y}$.

Darboux function

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 ${\displaystyle y=f(c)}$. 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: $${\displaystyle x\mapsto \{\begin{array}{ll}\sin (1/x)& \text{for }x\ne 0,\\ 0& \text{for }x=0.\end{array}}$$

By Darboux's theorem, the derivative of any differentiable function is a Darboux function. In particular, the derivative of the function ${\displaystyle x\mapsto x^2\sin (1/x)}$ 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 ${\displaystyle f(x)=\sum _{k=1}^{\mathrm{\infty }}\frac{(-1{)}^{x_k}}{k}}$ 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 ${\displaystyle F_{\sigma }}$ set. Conversely, every meagre ${\displaystyle F_{\sigma }}$ 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 ${\displaystyle f}$, 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 ${\displaystyle M_0,\dots ,M_5}$ 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 ${\displaystyle M_3}$.
• 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 ${\displaystyle M_4}$.

Intuitively, if ${\displaystyle f=F^{\prime }}$ 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 ${\displaystyle \mathrm{dist}(x,I_n)\to 0}$ and ${\displaystyle \lambda (I_n\cap E)=0}$, satisfies $${\displaystyle \frac{\lambda (I_n)}{\mathrm{dist}(x,I_n)}\to 0,}$$ 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.