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 2 of 6: The nonzero origin forces a constant

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In plain words

Once every value has floor one, the second variable set to zero makes the equation copy the origin value everywhere.

f(x)=f(0)=c,\quad 1\le c<2

Detailed analysis

In the branch $f(0)\ne0$ , set $y=0$ in the original equation. Since $\lfloor f(0)\rfloor=1$ , this gives $f(0)=f(x)\lfloor f(0)\rfloor=f(x)$ for every $x$ . Thus $f$ is constant, say $f(x)=c$ , and the earlier bound gives $1\le c<2$ .