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 other functions 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, $f\colon I\to \mathbb{R}$ a real-valued differentiable function. Then $f'$ has the intermediate value property: If $a$ and $b$ are points in $I$ with $a<b$ , then for every $y$ between $f'(a)$ and $f'(b)$ , there exists an $x$ in $(a,b)$ such that $f'(x)=y$ .^[1]^[2]^[3]

Proofs

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

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

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

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

Likewise, because $\varphi '(b)=f'(b)-y<0$ , we know $\varphi$ cannot attain its maximum value at $b$ .

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

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

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

Thus, for $t\in (a,b)$ we have $a\leq \alpha (t)<\beta (t)\leq b$ . 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, $g(t)\rightarrow {f}'(a)$ when $t\rightarrow a$ and $g(t)\rightarrow {f}'(b)$ when $t\rightarrow b$ ; therefore, from the Intermediate Value Theorem, if $y\in ({f}'(a),{f}'(b))$ 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 ${f}'(x)=g(t_{0})$ . Hence, ${f}'(x)=y$ .

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.^[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 function

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 $x\mapsto x^{2}\sin(1/x)$ 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.^[5] 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. Such functions exist and are Darboux but nowhere continuous.^[4]

Notes

1 2 Apostol, Tom M.: Mathematical Analysis: A Modern Approach to Advanced Calculus, 2nd edition, Addison-Wesley Longman, Inc. (1974), page 112.
1 2 Olsen, Lars: A New Proof of Darboux's Theorem, Vol. 111, No. 8 (Oct., 2004) (pp. 713–715), The American Mathematical Monthly
↑ Rudin, Walter: Principles of Mathematical Analysis, 3rd edition, MacGraw-Hill, Inc. (1976), page 108
1 2 Ciesielski, Krzysztof (1997). Set theory for the working mathematician. London Mathematical Society Student Texts. 39. Cambridge: Cambridge University Press. pp. 106–111. ISBN 0-521-59441-3. Zbl 0938.03067.
↑ Bruckner, Andrew M: Differentiation of real functions, 2 ed, page 6, American Mathematical Society, 1994

External links

This article incorporates material from Darboux's theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
Hazewinkel, Michiel, ed. (2001) [1994], "Darboux theorem", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

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[Apostol1974-1] 1 2 Apostol, Tom M.: Mathematical Analysis: A Modern Approach to Advanced Calculus, 2nd edition, Addison-Wesley Longman, Inc. (1974), page 112.

[Olsen2004-2] 1 2 Olsen, Lars: A New Proof of Darboux's Theorem, Vol. 111, No. 8 (Oct., 2004) (pp. 713–715), The American Mathematical Monthly

[Rudin1976-3] Rudin, Walter: Principles of Mathematical Analysis, 3rd edition, MacGraw-Hill, Inc. (1976), page 108

[Cie-4] 1 2 Ciesielski, Krzysztof (1997). Set theory for the working mathematician. London Mathematical Society Student Texts. 39. Cambridge: Cambridge University Press. pp. 106–111. ISBN 0-521-59441-3. Zbl 0938.03067.

[5] Bruckner, Andrew M: Differentiation of real functions, 2 ed, page 6, American Mathematical Society, 1994