How to notate this kind of function?

Question

I have a function $f$ whose domain is $\mathbb{N}$ and range is a set of another functions. The domain of $f(n)$ is $\mathbb{N}_0 \cap [0,\frac{n}{2})$ and its codomain is $\mathbb{Z}$. How can I notate the codomain of $f$?

Answer 1

You can use either

$$f:\Bbb N\to\bigcup_{n\in\Bbb N}{^{n}\Bbb Z}$$

or

$$f:\Bbb N\to\bigcup_{n\in\Bbb N}\Bbb Z^{n}\;,$$

where $n=\{0,1,\ldots,n-1\}$.

In general ${{^A}B}$ and $B^A$ are used for the set of functions from $A$ to $B$.

The fact that the domain of $f(n)$ is a function of $n$ pretty much has to be specified separately.