Let $U\subset\mathbb R^m$ be a connected open subset and $f:U\to U$ a function of class $C^k, (k\ge 1)$. Suppose $f\circ f=f$ and $M=f(U)$. I need a hint to prove that $f$ has constant rank in a neighborhood of $M$. I tried to use the chain rule, but I didn't make any progress.
Using this fact can I say $M$ is a manifold of class $C^k$?