How to prove a function is periodic?

Question

$$f(x)=\begin{cases}1&\text{if }2n-1<x<2n,\\0&\text{if }2n<x<2n+1. \end{cases} $$ Is this function is periodic or not? How can I prove it?

Answer 1

The function is defined on $\mathbb R\setminus \mathbb Z$ and has period $2$. To show the latter note that if $2n-1<x<2n$ then $2n'-1<x-2<2n'$ with $n'=n-1$ and if $2n<x<2n+1$ then $2n'<x-2<2n'+1$ with $n'=n-1$.