Thread: Continuity Proof

Let f(x) be a function that is defined for all real x and is continuous on its domain.

Suppose that lim (x-> negative infinity) =infinity and lim (x-> infinity)= negative infinity.

Show that there is a value of x such that f(f(x)) = x.



My intuition says that I should use the IVT, but I'm not sure how...any suggestions?

Consider the continuous function . Since as there must be some point where (you should prove this). Similarly, there must be some point where . Then since , there must be some such that by the IVT. Then clearly

So you were right about the intermediate value theorem!