Question 1
Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ such that for all $x,y \in \mathbb{R}$, $$
f(\left\lfloor x\right\rfloor y) = f(x)\left\lfloor f(y)\right\rfloor.
$$ Here $\lfloor z\rfloor$ denotes the greatest integer less than or equal to $z$.
f(\left\lfloor x\right\rfloor y) = f(x)\left\lfloor f(y)\right\rfloor.
$$ Here $\lfloor z\rfloor$ denotes the greatest integer less than or equal to $z$.