= Continuous maps between cofinite and tail topologies
For a map $f:\mathbb N\to\mathbb N$ from the <cofinite topology> to the <tail topology on the natural numbers>, <continuity> is equivalent to every set $\{n:f(n)\ge k\}$ being empty or cofinite. If the image is unbounded, every such set is nonempty and cofinite, so $f(n)\to\infty$. If it is bounded, its maximum $M$ exists, and the inverse image of the tail at $M$ forces $f(n)=M$ eventually, with $f(n)\le M$ always. Conversely either condition makes every tail preimage empty or cofinite.
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