Question about the properties of iterated functions

[Shanghai46]

Yes that is true if the function is continu and monotone on that interval.

Notice a violation would imply there is another fixpoint in the interval but that violates the conditions.
So it must be true then if  the function is continu and monotone on that interval.

So basically if f(x) is continu and f'(x) is never 0 or infinite on that closed interval.

Notice monotone implies the function and its inverse have unique inverses on that interval.
The inverse of a continu and monotone function is continu and monotone too afterall.



regards

tommy1729

Is there any theorems about this property, or any ways to demonstrate it?