Let $\mathbb{N}$ have the co-finite topology, and let $\mathbb{R}$ have the usual topology. Then what functions from $\mathbb{N}$ to $\mathbb{R}$ are continuous?
I think the constant functions would be continuous. The range of any such function (not constant) has to be countable. One can pick an open interval around one of $f(x')$, $x' \in \mathbb{N}$, such that no other $f(x)$ for $x \in \mathbb{N}$ lies in that open interval. The inverse of this open set under $f$ would not be open in co-finite topology being a finite set. Thus constant functions are the only continuous functions.
I am not sure if my argument is correct.