# Darboux's theorem (analysis)

**Darboux's theorem** is a theorem in real analysis, named after Jean Gaston Darboux. It states that all functions that result from the differentiation of other functions have the **intermediate value property**: the image of an interval is also an interval.

When *f* is continuously differentiable (*f* in *C*^{1}([*a*,*b*])), this is a consequence of the intermediate value theorem. But even when *f′* is *not* continuous, Darboux's theorem places a severe restriction on what it can be.

## Darboux's theorem

Let be an open 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]}

## Proof

If equals or , then setting equal to or , respectively, works. Therefore, without loss of generality, we may assume that is strictly between and , and in particular that . Define a new function by

Since is continuous on the closed interval , its maximum value on that interval is attained, according to the extreme value theorem, at a point in that interval, i.e. at some . Because and , Fermat's theorem implies that neither nor can be a point, such as , at which attains a local maximum. Therefore, . Hence, again by Fermat's theorem, , i.e. .^{[1]}

Another proof based solely on the mean value theorem and the intermediate value theorem is due to Lars Olsen.^{[1]}

## Darboux function

A **Darboux function** is a real-valued function *f* which has the "intermediate value property": for any two values *a* and *b* in the domain of *f*, and any *y* between *f*(*a*) and *f*(*b*), there is some *c* between *a* and *b* with *f*(*c*) = *y*.^{[2]} By the intermediate value theorem, every continuous function 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 that is not continuous.

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 *f* on the real line can be written as the sum of two Darboux functions.^{[3]} 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.^{[2]}

## Notes

- ↑
^{1.0}^{1.1}^{1.2}Olsen, Lars:*A New Proof of Darboux's Theorem*, Vol. 111, No. 8 (Oct., 2004) (pp. 713–715), The American Mathematical Monthly - ↑
^{2.0}^{2.1}{{#invoke:citation/CS1|citation |CitationClass=book }} - ↑ Bruckner, Andrew M:
*Differentiation of real functions*, 2 ed, page 6, American Mathematical Society, 1994

## External links

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