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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 another function has the **intermediate value property**: the image of an interval is also an interval.

When *ƒ* is continuously differentiable (*ƒ* in *C*^{1}([*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.

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 in such that .^{[1]}^{[2]}^{[3]}

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

If equals or , then setting equal to or , 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 on 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 .)

Likewise, because , we know cannot attain its maximum value at .

Therefore, must attain its maximum value at some point . Hence, by Fermat's theorem, , i.e. .

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

Define . For define and . And for define and .

Thus, for we have . Now, define with . is continuous in .

Furthermore, when and when ; therefore, from the Intermediate Value Theorem, if then, there exists such that . Let's fix .

From the Mean Value Theorem, there exists a point such that . Hence, .

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:

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.^{[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]}

- ^
^{a}^{b}Apostol, Tom M.: Mathematical Analysis: A Modern Approach to Advanced Calculus, 2nd edition, Addison-Wesley Longman, Inc. (1974), page 112. - ^
^{a}^{b}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- ^
^{a}^{b}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

*This article incorporates material from Darboux's theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*- "Darboux theorem",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994]