Prove that there exists no function such that and

(1)

**Solution**

Since forall this means that the function is strictly increasing in . Hence forall which in turn implies forall . Thus, . On the other hand , since is strictly increasing we get

(2)

From the first equation we have successively:

Thus,

(3)

Using equation we conclude that

(4)

Using equation we get that

Finally,

which is an obscurity. Hence , there is no such function!