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 5 of 6: The remaining branch is impossible

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In plain words

Invariance under the floor map makes every point in the first unit interval equal to the origin, and one carefully chosen pair then contradicts the value at one.

f(1)=f(2)\lfloor f(1/2)\rfloor=0

Detailed analysis

It remains to rule out $1\le f(1)<2$ . In this branch $\lfloor f(1)\rfloor=1$ , so setting $y=1$ gives $f(\lfloor x\rfloor)=f(x)$ for every $x$ . For $0\le x<1$ , this implies $f(x)=f(0)=0$ , hence $f(1/2)=0$ . Now choose $x=2$ and $y=1/2$ ; the equation gives $f(1)=f(2)\lfloor f(1/2)\rfloor=0$ , contradicting $1\le f(1)$ .