Question 43
Find all monotonic functions $f : \mathbb{R} \to \mathbb{R}$ satisfying the condition: for every real number $x$ and every natural number $n$
$$
\left| \sum_{i=1}^{n} i (f(x + i + 1) - f^2(x + i)) \right| \lt C
$$
where $C \gt 0$ is independent of $x$ and $f^2(x) = f(f(x))$.
$$
\left| \sum_{i=1}^{n} i (f(x + i + 1) - f^2(x + i)) \right| \lt C
$$
where $C \gt 0$ is independent of $x$ and $f^2(x) = f(f(x))$.