Let $f$ be a continuous function whose domain includes $[0,1]$, such that $0 \le f(x) \le 1$ for all $x \in [0,1]$, and such that $f(f(x)) = 1$ for all $x \in [0,1]$. Prove that $\int_0^1 f(x)\,dx > \frac34$.
Here's all that I have, from the Mean Value Theorem, we have some $c\in[0,1]$, and $a$, such that $$a=f(c)=\int_0^1 f(x)dx.$$ By the Extreme Value Theorem, there exist some $m$, $n\in[0,1]$ such that $$f(m)\ge f(x)\ge f(n).$$ I'm stuck here. Is this the right approach? Where do I go from here?
I also got to know what the very fact that $f(f(x))=1$ shows that there is some $x$ such that $f(x)=1$ because the range of $f(x)$ is the domain of $f(x)$ (which I'm still trying to understand; I know what it means, I'm just trying to take it in).