(Apologies for the vague way the question is stated.)
We define the function like this:
(1) $f(x) = x$ for $0\leq x \leq a$ (for some $a \geq 15 \in \mathbb{R}$) $\land$ $f(x) = a$ if $ x > a$ . For $x<0$, it doesn't really matter what $f(x)$ looks like.
Has someone studied this type of function? Does it have a 'nicer' definition than the one stated above? Any references on the subject? It is not absolutely necessary for the function to have all of the above characteristics. If you know a function that is similar to it, please let me know.
Motivation: I'm looking for a function that assigns a constant to 'infinite values' of x and does not 'deform' finite values (too much). So the function should satisfy $f(\zeta(1))=a$ for some finite value of $a \in \mathbb{R}$ and $f(3)=3$, for example. It doesn't really matter whether $f(b)=b$ or any multiple of b or that $f(x)$ is some finite polynomial, as long as it's 'easy' to find the value of $f(b)$.
Thanks,
Max Muller