converse of Darboux’s theorem (analysis) is not true
Darboux’ theorem says that, if has an antiderivative, than has to satisfy the intermediate value property, namely, for any , for any number with or , there exists a such that . With this theorem, we understand that if does not satisfy the intermediate value property, then no function satisfies on .
Now, we will give an example to show that the converse is not true, i.e., a function that satisfies the intermediate value property might still have no antiderivative.
First let us see that satisfies the intermediate value property. Let . If or , the property is satisfied, since is continuous on and . If , we have and . Let be between and . Let for some large enough such that . Then , and since is continuous on , we must have a with .
Assume, for a contradiction that there exists a differentiable function such that on . Then consider the function which is defined on . We have on , and since it is a an open connected set, we must have on for some . But then, we have
which contradicts the differentiability of at .
|Title||converse of Darboux’s theorem (analysis) is not true|
|Date of creation||2013-03-22 17:33:51|
|Last modified on||2013-03-22 17:33:51|
|Last modified by||Gorkem (3644)|