If a function series converges pointwise to a continuous function f(x), isn't it automatically uniformly convergent?
I can see how it doesn't have to be uniform when f(x) isn't continuous.
For example, fn(x) = x^n will converge to a function f(x) that is equal to 0 when x<1 and equal to 1 when x=1. The convergence isn't uniform, but f(x) isn't continuous.