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.

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- Mar 6th 2009, 07:45 AMChandru1Open set
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. - Mar 6th 2009, 09:24 AMPlato
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.