Let $\tau$ be the topology on $\mathbb{N}$ consisting of the empty set and all sets $X \subset \mathbb{N}$ such that $\mathbb{N} \backslash X$ is finite. Let $\sigma$ be the usual topology on $\mathbb{R}$, and let $\rho$ be the topology on $\mathbb{R}$ consisting of the empty set and all sets of the form $(x, \infty)$ for some real $x$.

(i) Prove that all continuous functions $f:(\mathbb{N}, \tau) \rightarrow(\mathbb{R}, \sigma)$ are constant.

(ii) Give an example with proof of a non-constant function $f:(\mathbb{N}, \tau) \rightarrow(\mathbb{R}, \rho)$ that is continuous.