Prove that $f(x) = x$ is the unique solution to the following functional equation

Question

Let $f : [0 , \infty) \to \mathbb{R},$ continuous at $ x_0 = 0$ satisfying $$f(3x) - 2x = f(x)$$ Prove that $f(x) = x$.

Answer 1

$f(3x)=2x+f(x) =2x+2x/3+f(x/3) =2x+2x/3+2x/9+f(x/9)$ $2+2/3+2/9+...=2*1/(1-1/3)=3$. By continued on 0 $f(3x)=3x$