Darboux's theorem (analysis)

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 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. Let be a closed interval, be 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. 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.