# Math Help - does such a function exist?

1. ## does such a function exist?

I have a function h with the property
$
\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:

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

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

$

2. For the first question, let h(x)=-1/x (-1<x<0) .

3. I think, for the second question, the answer is NO.

4. but -1/x is not monotonic on the whole real line
for instance $-1<2 \ and \ -1/-1 = 1>-1/2$

I think that the second is true for for instance $h^{-1}=-x$

is this right?

5. Originally Posted by hanns
but -1/x is not monotonic on the whole real line
for instance $-1<2 \ and \ -1/-1 = 1>-1/2$