Problem 5

Find all functions

f

mapping the non-negative reals onto the non-negative reals such that

f(xf(y))f(y)=f(x+y)

for all non-negative reals

x,y

, with

f(2)=0

and

f(x)\ne0

for every

0\le x<2

Step 1 of 4: Propagate the zero

In plain words

The special value at $2$ immediately controls the entire tail.

y=2\Longrightarrow f(x+2)=0\Longrightarrow f(t)=0\ (t\ge2)

Detailed analysis

Set $y=2$ in $f(xf(y))f(y)=f(x+y)$ . Since $f(2)=0$ , $f(x+2)=0$ for all $x\ge0$ , hence $f(t)=0$ whenever $t\ge2$ .