Suppose that $X$ is a compact Hausdorf space, and that every continuous function on $X$ has finite range. How do I conclude that $X$ is a finite set, hence with discrete topology? So far, I have managed to use Urysohn to show that the closed maximal connected component decomposition of $X$ (as opposed to the path component decomposition) is one where every component is a 1 point set.
This question arose from trying to solve the question that every C* algebra with all normal elements having finite spectra are finite dimensional. A more direct solution to this general question would also suffice, because actually implies the special case I've reduced it to. By taking a unitization, and then a MASA in the unitization, one gets to the case I've gotten to.
Also, a topological question that is in the same setup but handles a related operator algebras question is this: if a net $u_i$ and $u$ in $X$ has the property that for all continuous $f \in C(X)$ one has that $f(u_i)$ converges to $f(u)$ then must $u_i$ converge to $u$ in topology? This is for showing that the gelfand representation of a $C(X)$ space is itself. I'd like this fact for $X$ locally compact Hausdorf, actually.