2 real fixpoints ... 3 iterations ?

[jaydfox]

regular iteration at fp 1 is valid at zone 1&2 and regular iteration at fp 2 is valid at zone 2&3.

no , you assume regular iteration at fp1 touches fp 2 , what if it doesnt ?

But wouldn't that necessitate another fixed point between fp1 and fp2? Why else would regular iteration from fp1 not reach fp2? By your original stipulation, the function is strictly increasing (except, of course, at the fixed points). If it does not get from fp1 to fp2, there must be some point (which we can find by taking the limit of infinite iterations) which is a fixed point.