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 [math]\displaystyle{ I }[/math] be a closed interval, [math]\displaystyle{ f\colon I\to \R }[/math] be a real-valued differentiable function. Then [math]\displaystyle{ f' }[/math] has the intermediate value property: If [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math] are points in [math]\displaystyle{ I }[/math] with [math]\displaystyle{ a\lt b }[/math], then for every [math]\displaystyle{ y }[/math] between [math]\displaystyle{ f'(a) }[/math] and [math]\displaystyle{ f'(b) }[/math], there exists an [math]\displaystyle{ x }[/math] in [math]\displaystyle{ [a,b] }[/math] such that [math]\displaystyle{ f'(x)=y }[/math].

Proofs

Proof 1. The first proof is based on the extreme value theorem.

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

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

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

Likewise, because [math]\displaystyle{ \varphi'(b)=f'(b)-y\lt 0 }[/math], we know [math]\displaystyle{ \varphi }[/math] cannot attain its maximum value at [math]\displaystyle{ b }[/math].

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

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

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

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

Furthermore, [math]\displaystyle{ g(t) \rightarrow {f}' (a) }[/math] when [math]\displaystyle{ t \rightarrow a }[/math] and [math]\displaystyle{ g(t) \rightarrow {f}' (b) }[/math] when [math]\displaystyle{ t \rightarrow b }[/math]; therefore, from the Intermediate Value Theorem, if [math]\displaystyle{ y \in ({f}' (a), {f}' (b)) }[/math] then, there exists [math]\displaystyle{ t_0 \in (a, b) }[/math] such that [math]\displaystyle{ g(t_0) = y }[/math]. Let's fix [math]\displaystyle{ t_0 }[/math].

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

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:

[math]\displaystyle{ x \mapsto \begin{cases}\sin(1/x) & \text{for } x\ne 0, \\ 0 &\text{for } x=0. \end{cases} }[/math]

By Darboux's theorem, the derivative of any differentiable function is a Darboux function. In particular, the derivative of the function [math]\displaystyle{ x \mapsto x^2\sin(1/x) }[/math] 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