Problem:
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 $\int_0^1 f(x)\,dx > \frac34$.
Notes: I am working on it right now so I will update this with my work.
I believe there is another question just like this one already answered.
I just want hints to try to figure it out on my own.
Hint: Since $f$ is continuous, it doesn't skip intermediate values in the range $0\le f(x)\le 1.$ Thus, since $f(f(x))=1$ over that range, it follows that $f$ is constant over $[0,1].$
Can you finish off?