
Continuity Proof
Let f(x) be a function that is defined 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 $\displaystyle f(x)x$. Since $\displaystyle f(x)\rightarrow \infty$ as $\displaystyle x \rightarrow \infty$ there must be some point $\displaystyle y\in\mathbb{R}$ where $\displaystyle f(y)>y$ (you should prove this). Similarly, there must be some point $\displaystyle z\in\mathbb{R}$ where $\displaystyle f(z)<z$. Then since $\displaystyle 0 \in [f(z)z,f(y)y]$, there must be some $\displaystyle x\in[z,y]$ such that $\displaystyle f(x)x = 0$ by the IVT. Then clearly $\displaystyle f(f(x)) = f(x) = x.$
So you were right about the intermediate value theorem!