I tried but failed to find some references about the following iterative functional equation: find all functions $f \colon \mathbb{R} \to \mathbb{R}$ such that $f \circ f = f$, that is $f(f(x))=f(x)$ for all real numbers $x$.
There are some obvious continuous solutions: the constant functions, $f(x)=x$, $f(x) = \pm \vert x \vert$.
There are also very pathological solutions. If $(e_i)_{i \in I}$ is a Hamel basis of $\mathbb{R}$ seen as a $\mathbb Q$-vector space with $e_{i_0}=1$ for a given $i_0 \in I$, then the projection onto $\mathbb Qe_{i_0}$ is a nowhere continuous solution.
I would be very interested if there is a characterization of all continuous solutions, and eventually with "weaker" conditions (monotony, continuity just at one point, etc).
Any help would be appreciated, thanks !
