analysis - $f(f(x))=f(x)$ question

I am wondering what is the class of functions $f: \mathbb{R}\rightarrow\mathbb{R}$ such that $f(f(x))=f(x)$?

I think it should be:

1. Constant Value functions


2. the identity function


3. absolute value function $|x|$

But I don't know if this is right or how to show it rigorously.

Any suggestions?