New posts in Extreme Value Theorem

Given $f(x)$ is continuous on $[0,1]$ and $f(f(x))=1$ for $x\in[0,1]$. Prove that $\int_0^1 f(x)\,dx > \frac34$.

Trouble understanding how the Transfer Principle is applied for the Extreme Value theorem.

$C^1$ function on compact set is Lipschitz