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 5: Find the zero region

In plain words

Substituting the known zero $f(2)$ into the functional equation propagates that zero to every argument at least $2$ .

f(x+2)=f(xf(2))f(2)=0

Detailed analysis

Putting $y=2$ gives $f(x+2)=f(xf(2))f(2)=0$ for every $x\ge0$ . Hence $f(t)=0$ for every $t\ge2$ .