Not continuous function $f: \mathbb{Z} \to \mathbb{Z}$

Question

Let $\mathbb{Z}$ denote the integers endowed with the cofinite topology. Exhibit an example of a function $f: \mathbb{Z} \to \mathbb{Z}$ which is not continuous.

I really need help with this problem; I'm unsure where to start.

Answer 1

The function $f:\Bbb Z\to\Bbb Z $ defined by $f (x)=(-1)^x $ is not continuous because $f^{-1}\{1\}=2\Bbb Z $, $\{1\} $ is closed but $2\Bbb Z $ is not closed.