Problem 1

Find all functions

f:\mathbb R\to\mathbb R

such that, for all real numbers

x,y

f(\lfloor x\rfloor y)=f(x)\lfloor f(y)\rfloor

Step 4 of 6: The zero value at one gives the zero function

In plain words

Putting one in the first variable removes the floor multiplier from the left, so a zero value at one annihilates every output.

f(y)=0\quad\text{for all }y\in\mathbb R

Detailed analysis

If $f(1)=0$ , put $x=1$ and let $y$ be arbitrary. The equation becomes $f(y)=f(1)\lfloor f(y)\rfloor=0$ , so $f$ is identically zero.