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 3 of 6: Split the zero-origin branch at one

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In plain words

The same equation at the input one leaves only a zero value or a value whose floor is one.

f(1)=f(1)\lfloor f(1)\rfloor\quad\Longrightarrow\quad f(1)=0\ \text{or}\ 1\le f(1)<2

Detailed analysis

Assume now $f(0)=0$ . Taking $x=y=1$ gives $f(1)=f(1)\lfloor f(1)\rfloor$ . Hence either $f(1)=0$ , or $\lfloor f(1)\rfloor=1$ , equivalently $1\le f(1)<2$ .