Prove that piecewise function is well-defined

Question

How can I rigorously prove that the function $f: [1, \infty)\mapsto \mathbb{R}$ \begin{equation} f(x)= \begin{cases} x+1\hspace{0.2mm} &\text{if }x\in [2k-1, 2k) \\ x-1\hspace{0.2mm} &\text{if }x\in [2k, 2k+1) \\ \end{cases} \end{equation} where $k=1, 2, 3, …$, is indeed well-defined, that is, all the intervals on the right are pairwise disjoint and their union is $[1, \infty)$?

Answer 1

By noting that all intervals are of the form $[n,n+1)$ (with $n\in\mathbb Z$) and that all the $n$'s are distinct