A continuous function on closed bounded X subset of R is uniformly convergent. a uniformly convergent sequence of functions must converge to a uniformly continuous function.
If that is your meaning of "uniform limit", then the limit function is continuous.
A continuous function on closed bounded X subset of R is uniformly continuous.
This is also true because closed bounded X subset of R are compact.
Last edited by Plato; October 25th 2008 at 04:33 AM.