Darboux's theorem (analysis) explained

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\colon I\to \R be 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 \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 = (1)/(2) (a + b) .For a \leq t \leq c, define \alpha (t) = a and \beta (t) = 2 t - a .And for c \leq t \leq b, define \alpha (t) = 2 t - 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) = ((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) → {f}' (a) when t → a and g(t) → {f}' (b) when t → 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 topologist's sine curve function:

x \mapsto \begin{cases}\sin( 1 /x) & for x\ne 0 , \ 0 & 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. The Conway base 13 function is again an example.^[4]

Notes and References

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