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

[Catullus]

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 \(n\in\mathbb Z\).

Log(sqrt(2),sqrt(2)^x+sqrt(2))-1 is a smooth approximation of max(x,1)-1.