1. ## 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?

2. Consider the continuous function $f(x)-x$. Since $f(x)\rightarrow \infty$ as $x \rightarrow -\infty$ there must be some point $y\in\mathbb{R}$ where $f(y)>y$ (you should prove this). Similarly, there must be some point $z\in\mathbb{R}$ where $f(z). Then since $0 \in [f(z)-z,f(y)-y]$, there must be some $x\in[z,y]$ such that $f(x)-x = 0$ by the IVT. Then clearly $f(f(x)) = f(x) = x.$

So you were right about the intermediate value theorem!