IMO 2010 Problem 1

Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds $$f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor$$ where $\left\lfloor a\right\rfloor$ is greatest integer not greater than $a.$ Proposed by Pierre Bornsztein, France