TITLE: If $f(x)\leq f(f(x))$ for all $x$, is $x\leq f(x)$?
QUESTION [15 upvotes]: If I have $f(x)\leq f(f(x))$ for all real $x$, can I deduce $x\leq f(x)$?

Thank you.

REPLY [5 votes]: If $f$ satisfies the above conditions, then:

For all $x \in Im(f)$ we have $x \leq f(x)$. In particular, if $f(x)$ is onto then, the problem is true.
To see this, let $y$ be so that $x=f(y)$, then $f(y) \leq f(f(y)) \Rightarrow x \leq f(x)$.
There exists non onto functions which satiesfy the above condition for which, $x \leq f(x)$ if and only if $x \in Im(f)$. For example $f(x)=\frac{x-|x|}{2}$.