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 pointwise continuity vs uniform continuity, but does not expound upon the analogous issue of pointwise vs uniform convergence.
