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 1 of 6: Substitute zero for the first variable

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In plain words

The floor of zero makes the left side independent of the second variable, so the value at the origin controls every floor on the right.

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

Detailed analysis

Putting $x=0$ into the equation gives $f(0)=f(0)\lfloor f(y)\rfloor$ for every real $y$ . If $f(0)\ne0$ , division by $f(0)$ yields $\lfloor f(y)\rfloor=1$ for every input, so $1\le f(y)<2$ for every $y$ . Otherwise, we are in the case $f(0)=0$ .