Non-trivial extension of max(n,1)-1 to the reals and its iteration.

[tommy1729]

Your function is equal to (x + abs(x))/2.

abs(x) can be written as sgn(x) x.

sgn(x) is well approximated by tanh(100x).

This gives that your function is very close to

(x+tanh(100x) x)/2

The problem with your function is that it has all positive integers as fixpoints.
Too many fixpoints to have half-iterates valid everywhere.

Im not sure if you want an interpolation or an approximation like I just gave.

Also the reason you get little response is probably because your mainly asking " what makes this question more intresting " ?
If you know what I mean.

Asking what properties to look for or asking what questions to ask is similar.

You have to decide what you want to do , want to see solved or what properties you desire.

Otherwise it sounds weird. Kinda like asking for " a special integer ".

Math is like driving a car without a map.
You dont know where you will end up. But if you want to end up somewhere you have too start , stop and drive.

I hope my metaphor is understood.

I assume you are still young. You dont have to tell me about your age but I suspect it.
Hope you dont mind me saying.

regards

tommy1729