A map is called open if is open for every open
subset A of R. Show that every continuous open map of R into
itself is monotonic.
Because this function is continuous then on any interval it has a high point and a low point: .
Suppose that then which is contradictory to being open.
A similar argument leads to the conclusion that the maximum and minimum must happen at the endpoints of any closed interval.
Suppose that there are points .
But that would mean that the maximum on the interval is not at endpoint.
The same contradiction would follow if .
In other words must be monotonic.