I have a couple of questions on sequences of functions to help me understand the concept better.
If you have a function f, what must be required of the function to be able to build a sequence of convergent functions. Is it just that the sequence has to converge to f?
If you have a differentiable function f, then will the sequence of functions that converges to f all be continuous (and differentiable)? And if they are all continuous then will the sequence be pointwise convergent?
(I have seen examples of sequences of functions that were all continuous but the sequence wasn't uniformly continuous so I know continuity does not imply uniform continuity, but what about pointwise?)