I have a function h with the property

$\displaystyle

\forall x,y \in R,\ x\leq y \rightarrow h\left(x\right) \leq h\left(y\right)

$

Now A is a nonempty set which has a supremum

Are there functions such that:

$\displaystyle

1. h\left(A\right) \ has \ not\ a\ supremum $

$\displaystyle

2. h^{-1}\left(A\right) \ has\ not\ a\ supremum

$