= Solution
There is a small defect in the printed definition: the listed family omits $\mathbb R$ and is not a <topology>, since $\bigcup_{n\geq1}(-n,\infty)=\mathbb R$. We use the intended <upper-ray topology on the real line>, obtained by adjoining $\mathbb R$ to that family.
Take \b[$f(n)=n$]. For every real $a$,
$$
f^{-1}((a,\infty))=\{n\in\mathbb N:n>a\}
$$
has finite complement and is therefore open in the <cofinite topology>. The inverse images of the empty set and the whole line are also open. Hence this nonconstant function is a <continuous map> for the corrected <upper-ray topology on the real line>. The same inverse-image calculation verifies the condition for every ray literally listed in the question.
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