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 .
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.
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 .
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. 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. 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.