Ok, that makes sense ( I assume you mean 1 not 0). In regards to the function I discussed, I still must describe the sets upon which f converges uniformly in terms of closed or bounded intervals, correct? I still would not be able to say that f converges uniformly on any open interval, even though every point of A is a point of B and visa versa.
If this is in fact the case I assume it comes from the nonsensicalness of discussing uniform convergence on single points; instead it must be discussed with respect to sets. Rudin makes this remark explicit with regard to point-wise continuity vs uniform continuity, but does not expound upon the analogous issue of point-wise vs uniform convergence.