Consider the function $f : x \mapsto \lfloor x \cos(x) \rfloor$ on the real numbers.
I would like to know whether the functions $f_n := \underbrace{f \circ ... \circ f}_{n \text{ times }}$ (with $n \geq 1$) are linearly independent over $\Bbb R$ (in the real vector space $\Bbb R^{\Bbb R}$).
The answers to my previous questions cannot be applied, since $f$ is cannot be extended to a holomorphic function, and $f$ is not bounded. I don't even know how to show that $f=f_1$ and $f_2$ are linearly independent over $\Bbb R$. Any hint would be appreciated.