Originally Posted by

**emakarov** It helps to draw a graph: first of $\displaystyle x/2$, then $\displaystyle \lfloor x/2\rfloor$, then $\displaystyle \lfloor x/2\rfloor/2$, and finally $\displaystyle \lfloor\lfloor x/2\rfloor/2\rfloor$.

The remainder is not often used for the division of real numbers. I believe it is sufficient to consider two cases: when the function argument has the form $\displaystyle 4n+x$ and $\displaystyle 4n+2+x$ where $\displaystyle n\in\mathbb{Z}$ and $\displaystyle 0\le x<2$. Then, obviously, $\displaystyle 0\le x/2<1$ and $\displaystyle 0\le x/4<1/2$.