If the continuous function $f:\mathbb{R} \to \mathbb{R}$ has the following property?

Question

If the continuous function $f:\mathbb{R} \to \mathbb{R}$ satisfy $f \circ f \circ f(x)=x$, then f(x)=x?

I have proved

1.$f$ is bijection ,so $\exists x_0, f(x_0)=0$ 2.if a<x_0<b, then f(a)f(b)<0

3.$f$ is monotone

but I'm not sure what to do after this.

Answer 1

Since $f$ is a continuous bijection it is strictly monotonic. Suppose it is strictly decreasing. Then $x<y$ implies $f(f(x)) >f(f(y))$ and $f(f(f(x))) >f(f(f(y)))$ or $x>y$, a contradiction. So $f$ is is strictly increasing.

Now let $f(x) >x$. Then $f(f(x)) >f(x)$ and $x=f(f((f(x)))>f(f(x)>x$, a contradiction. Similarly, $f(x) <x$ leads to a contradiction. Hence $f(x)=x$ for all $x$.