Continuous order preserving function

Question

Give an example of continuous function which is order preserving and its inverse is not order preserving ?

Answer 1

If $f$ is an order-preserving function on a totally-ordered set, and $f$ has an inverse, then $f^{-1}$ must be order-preserving. Perhaps you're dealing with a partially ordered set. Then you could have cases where $x$ and $y$ are incomparable, but $f(x)$ and $f(y)$ are comparable.

Answer 2

I think I found the solution let X be finite has more than one point set define a bijective function f:X to X where X as domain has discrete topology and X as co-domain has indiscrete topology Now f it is clear continuous hence its order preserving(by Lemma2.2 J.P. MAY page 3) but its inverse is not continuous and hence its not order preserving ....:)