Prove that an infinite set is semidecidible
Here is a rough sketch.

If $M$ recognizes $C$, we can exploit $M$ to craft $N$ which enumerates the language of $M$, writing all its words on an output tape, clearly separated, in infinite time. For that, it suffices to exploit dovetailing to generate all the pairs $(w,k)$, where $w$ is a word and $k$ a natural. For each such pair, we run $M$ on $w$ for $k$ steps -- if $M$ accepts, we add $w$ to the output tape, otherwise we try the next pair.

Exploiting $N$ is then possible to define a total computable $f$ whose image is $C$.



For the other direction, given a total computable $f$, we can craft $M$ such that on input $w$ compares it to $f(0),f(1),f(2),\ldots$. In this way we can make $M$ recognize the image of $f$, i.e. $C$.