Determination of $f(x)$

Question

A continuous function $f$ satisfies $f(x+f(x))=f(x)$ for all $x\in[0,1]$. prove that $f(x)$ is constant.

$f(f(0))=f(0)$. I am unable to proceed further.

Answer 1

Put $f(x)=0$ when $x\in\mathbb R_+\cup\{0\}$ and put $f(x)=x$ when $x\in\mathbb R_-$. This is not constant and satisfy youR question.