Školjka

%V0 Let $\mathbb{R}$ be the set of real numbers. Does there exist a function $f: \mathbb{R} \mapsto \mathbb{R}$ which simultaneously satisfies the following three conditions? (a) There is a positive number $M$ such that $\forall x:$ $- M \leq f(x) \leq M.$ (b) The value of $f(1)$ is $1$. (c) If $x \neq 0,$ then $$f \left(x + \frac {1}{x^2} \right) = f(x) + \left[ f \left(\frac {1}{x} \right) \right]^2$$