Intervals, Equalities and Functions - Part 1

Find the smallest possible output for $f : \mathbb{R} \rightarrow \mathbb{R}$ such that for all the following equality holds $f({\lfloor}x{\rfloor}y)$ $=$ $f(x){\lfloor}f(y){\rfloor}$

Details and Assumptions

${\lfloor}x{\rfloor}$ is the greatest integer less than $x$