Darboux's theorem (analysis)

Short description: All derivatives have the intermediate value property

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.

Darboux's theorem

Let $I$ be a closed interval, $f:I\to \mathbb{ℝ}$ 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 $f^{\prime }(x)=y$.

Proofs

Proof 1. 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 $\phi :I\to \mathbb{ℝ}$ 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 2. 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 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. 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:

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

By Darboux's theorem, the derivative of any differentiable function is a Darboux function. In particular, the derivative of the function $x\mapsto x^2\sin (1/x)$ 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.

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. The Conway base 13 function is again an example.

External links

• Hazewinkel, Michiel, ed. (2001), "Darboux theorem", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/d030190