Question 1
Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that the equality
$$
f([x] y)=f(x)[f(y)] .
$$
holds for all $x, y \in \mathbb{R}$. Here, by $[x]$ we denote the greatest integer not exceeding $x$.
$$
f([x] y)=f(x)[f(y)] .
$$
holds for all $x, y \in \mathbb{R}$. Here, by $[x]$ we denote the greatest integer not exceeding $x$.