Is the interior of a compact subet of C^0 empty?

This question is very similar to one I asked here: http://www.mathhelpforum.com/math-he...ons-empty.html

Let C0 be the set of continuous functions (from [a,b] to R) and a subspace of Cb, the set of bounded functions (from [a,b] to R), defined with the supnorm metric.

Let A be a compact subset of C0.

Is the Interior of A empty?

My first thought was that the proof is the same as in the link above (that is, yes it is empty since we can easily find a function g in any open ball around f in A where g is not continuous), but I'm worried this doesn't apply here because the subset A may only contain functions with certain properties, and so we don't know if g is in A.

Am I making any sense here? I think I successfully confused myself.