Here is the question; it may seem very simple, but it is difficult (at least for me).

Let $f(x)$ be a continuous function on $R$ that is strictly increasing, and suppose $g(x)=f(x)-x$ is a periodic function with period 1.

Prove that for all $x\in R$, $\lim_{n\to \infty}\frac{f^n(x)}{n}$ exists.

In an equivalent formulation, the dynamic system $(X,T)$ is quasi-regular, where $X=[0,1],T: x\mapsto x+\{g(x)\}$. I.e. the Birkhoff average $\frac{1}{N}\sum_{n\leq N}T^n(x)$ exists for all $x\in X$.

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