Complete Compact Linear Orders

From: William Elliot <marsh@xxxxxxxxxxxxxxxxxx>
Date: Wed, 1 Nov 2006 03:09:43 -0800

Let X,Y be complete linear orders
or equivalently, compact linear orders.

If f:X -> Y is ascending (not strictly increasing) and continuous
then for all A subset X,
inf f(A) = f(inf A), sup f(A) = f(sup A).

Conversely assume for all A subset X,
inf f(A) = f(inf A), sup f(A) = f(sup A).
It's easy to show f is ascending.
Is it possible to show f is continuous?
.

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