BALKAN-SL 2016 A3

Find all monotonic functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying the condition: for every real number $x$ and every natural number $n$

$$\left|\sum_{i=1}^{n} i\left(f(x+i+1)-f^{2}(x+i)\right)\right|<C$$

where $C>0$ is independent of $x$ and $f^{2}(x)=f(f(x))$.