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

[MphLee]

This will work, but is not really what I was looking for in my question. Back in the days my mind was really foggy on this problem because I wasn't able to phrase the real problem I had in mind.

Here indeed your solution  \((x-1)-\sin(x\pi)\) is correct.


But probably I was looking for somethign like a smooth approximation of \(T(x)\sim \max(x,1)-1\)  and then the family \(T_\theta=T(x)+\theta(x)\) for \(\theta(x+1)=\theta(x)\) and \(\theta(n)=0\) for each \(n\in\mathbb Z\).
So Tommy's solution fits better since \(\max(x,1)-1=\lim_{k\to\infty} \frac{(x-1)+(x-1){\rm tanh}(k(x-1))}{2} \)

Here Tommy's approximation for \(k=6\)


Here then \(T_\theta(x)=\frac{(x-1)+(x-1){\rm tanh}(k(x-1))}{2} + \lambda \sin(x\pi)\) for \(\lambda=0.2\)


After all this years I have to think more about this and see if those approximations can be useful for my endgame...
One day I'll post something about it.