Good evening;
A friend who is studying real analysis asked me for help in this problem, I would like any clue of solution or complete solution, I also accept bibliographical references.
Let $f: [0,1] \to [0,1]$ be a continuous function, bijective, strictly increasing monotonous, safisfying $f(0)=0$, $f(1)=1$ and $f(x)<x, \forall x \in (0,1)$. Let $\tau : [0,1] \to \mathbb{R}$ be a $C^{1}$ class function. For each points $x,y \in (0,1)$ consider the sequence:
$$ \Phi_{n}(x,y) = |\sum_{j=0}^{n-1}\tau(f^{j}(x))-\tau(f^{j}(y))|$$.
Prove that $\forall \epsilon > 0$ exist $\delta > 0$ such that: if $|x-y| < \delta$ then $\Phi_{n}(x,y)<\epsilon$, for all $n \in \mathbb{N}$.