Let us consider the following multivariate function: $f(x,y,z)$, where $x\in\mathbb{R},y\in\mathbb{R},z\in\mathbb{N}$ and $x\in[0,1],y\in[0,1],z=\{1,2,3,...k\}$.
For the function's domain I suggest the following notation: $$f(x,y,z):X\mapsto \mathbb{R}~, X=\{x\in\mathbb{R},y\in\mathbb{R},z\in\mathbb{N}\mid 0\leq x,y\leq 1,z=\overline{1,k}\}$$
How can I precisely write the function's domain is the above mentioned notation clear?
So, $f: [0;1]{\times}[0;1]{\times}([1;k]\cap\Bbb N)\mapsto \Bbb R$