If each $f_n$ is continuous on a set $S$, does $f_n$ converge pointwise to a function $f$ on $S$? If each $f_n$ is continuous on a set $S$, does $f_n$ converge pointwise to a function $f$ on $S$?I feel I am seriously misunderstanding something. Am I asking a vacuous question?

Vacuous, I'm not sure. But the answer, of course, is no. Even if your set $S$ is a closed interval in $\Bbb R$, consider $f_n(x)=nx$.