Continuous functions need hint

Question

We have continuous functions $f,g:[0;\infty)\longrightarrow[0;\infty)$ with the following properties:

1. $f(0)=g(0)=0$

2. $g(x)\neq0$, for any $x>0$

3. $f(x+g(f(x)))=f(x)$, for any $x$

Prove that $f(x)=0$ for any $x$. I need only a hint how to start. So far I've tried something with a sequence with positive terms and limit $0$. I think that somehow we have to get to: $g(f(x))=0$ for any x, from where the conclusion follows.

Answer 1

Let any $x_0>0$. Show that there exist $x_1$ with $0<x_1\leq x_0$ such that $x_1+g(f(x_1))=x_0$, using the intermediate value theorem. Now let $u>0$, define a sequence $x_n$ with $x_0=u$ and $0<x_{n+1}\leq x_n$, $x_{n+1}+g(f(x_{n+1}))=x_n$, and study $x_n$, and $f(x_n)$

Answer 2

I got it to work out by assuming there is an $x_0 > 0$ such that $f(x_0) \neq 0$. Then what does that tell you about $f(x_0 + g(f(x_0)))$? How can you use this?