Darboux's theorem (analysis)

In mathematics, Darboux's theorem is a theorem in real analysis, named after Jean Gaston Darboux. It states that every function that results from the differentiation of another function has the intermediate value property: the image of an interval is also an interval.

When ƒ is continuously differentiable (ƒ in C¹([a,b])), this is a consequence of the intermediate value theorem. But even when ƒ′ is not continuous, Darboux's theorem places a severe restriction on what it can be.

Let ${\displaystyle I}$ be a closed interval, ${\displaystyle f\colon I\to \mathbb {R} }$ 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]

Proof 1. 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. ${\displaystyle f'(x)=y}$.

Proof 2. The second proof is based on combining the mean value theorem and the intermediate value theorem.^[1][2]

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 ƒ which has the "intermediate value property": for any two values a and b in the domain of ƒ, and any y between ƒ(a) and ƒ(b), there is some c between a and b with ƒ(c) = y.^[4] 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 the Conway 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 ${\displaystyle f(x)=\sum \limits _{i=1}^{\infty }{\frac {(-1)^{x_{k}}}{k}}}$ if the series converges for that x and 0 if it does not.^[5]

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.^[6] 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. ^[4]

• 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]