If $f(f(x)) = x $ has at most 1 solution, then so does $f(x) = x$. 
Let f be defined on [0,1] and its values are between 0 and 1. If
  $f(f(x)) = x $ has at most 1 solution, then $f(x) = x$ has at most 1
  solution.

Please, give me a hint how to prove this.

This is a special case of a more general fact: if $f:A\to B$ and $g:B\to C$ are functions, and
$$g\circ f:A\to C$$
is injective, then $f$ is injective.
To see this, suppose that $f(x) = f(y)$ for some $x, y\in A$. Then $gf(x) = gf(y)$, so $x=y$.
