If $f(x)\leq f(f(x))$ for all $x$, is $x\leq f(x)$?
If I have $f(x)\leq f(f(x))$ for all real $x$, can I deduce $x\leq f(x)$?
Thank you.
Solution 1:
No. Suppose $f$ is a constant function.....
Solution 2:
For a slightly non-trivial example, $f(x) = - \vert x \vert$. We then have that $$f(f(x)) = - \vert f(x) \vert = -\vert-\vert x \vert \vert = - \vert x \vert = f(x)$$ However, $x \geq f(x)$ for all $x \in \mathbb{R}$ and in fact $x > f(x)$ for all $x \in \mathbb{R}^+$.