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 6 of 6: Verify the two families

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In plain words

The surviving functions satisfy the equation directly, so the case analysis is complete.

f(x)\equiv0\quad\text{or}\quad f(x)\equiv c\ (1\le c<2)

Detailed analysis

The function $f(x)\equiv0$ clearly satisfies the equation. If $f(x)\equiv c$ with $1\le c<2$ , then $\lfloor c\rfloor=1$ and both sides equal $c$ . Therefore the complete solution set is $f\equiv0$ and the constant functions $f\equiv c$ for $1\le c<2$ .