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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

be a closed interval,

a real-valued differentiable function. Then

has the intermediate value property: If

and

are points in

with

, then for every

between

and

, there exists an

such that

.^[1]^[2]^[3]

Proofs

equals

, then setting

equal to

, respectively, gives the desired result. Now assume that

is strictly between

and

, and in particular that

. Let

such that

. If it is the case that

we adjust our below proof, instead asserting that

has its minimum on

Since

is continuous on the closed interval

, the maximum value of

is attained at some point in

, according to the extreme value theorem.

Because

, we know

cannot attain its maximum value at

. (If it did, then

for all

, which implies

Therefore

must attain its maximum value at some point

. Hence, by Fermat's theorem,

, i.e.

Furthermore,

when

and

when

; therefore, from the Intermediate Value Theorem, if

then, there exists

such that

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.^[6] 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

By Darboux's theorem, the derivative of any differentiable function is a Darboux function. In particular, the derivative of the function

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.^[7] 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.^[6]

Notes

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. ^Apostol, Tom M.: Mathematical Analysis: A Modern Approach to Advanced Calculus, 2nd edition, Addison-Wesley Longman, Inc. (1974), page 112.
5. ^Olsen, Lars: A New Proof of Darboux's Theorem, Vol. 111, No. 8 (Oct., 2004) (pp. 713–715), The American Mathematical Monthly
6. ^¹{{cite book | last=Ciesielski | first=Krzysztof | title=Set theory for the working mathematician | zbl=0938.03067 | series=London Mathematical Society Student Texts | volume=39 | location=Cambridge | publisher=Cambridge University Press | year=1997 | isbn=0-521-59441-3 | pages=106–111 }}
7. ^Bruckner, Andrew M: Differentiation of real functions, 2 ed, page 6, American Mathematical Society, 1994