The main question is:
"How do derivatives/integrals interact with limits?"
EG, is the derivative of a limit the limit of the derivatives, etc.?
One can ask this pointwise, for derivatives, but it's clearer to do it
globally -- and in that case the right question is:

"Are D/I (the differentiation/integration -operators-) continuous
 in a given topology, and are the space of differentiable/integrable
 /continuous functions complete?"

"In particular, closed subsets of the space of -all- functions,
 which is complete in the pointwise topology?"

A natural topology is the -uniform- topology:
these turns out to be:
- just right for continuous
- stronger than needed for integration
- not strong enough for differentiation

...and the big picture is (-on a compact set!-):
$C^\infty < \dots < C^k < \dots < C^1 < C^0
  < L^\infty < \dots < L^p < \dots < L^1$
...and each of these spaces gets a progressively stronger "natural"
topology as you go to the left, such that they're all complete
and differentiation/integration are continuous.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
This topic gets intricate b/c there's lots of analysis you can prove,
esp. with differentiation.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

in summary:
- uniform limit of continuous is continuous
  (and is the pointwise limit: like, duh)
  it's the -right- topology (on compacts)

- uniform limit of integrable is integrable,
  [on compact set (more gen., finite measure)!
   otherwise have 1/x \chi_{[1,n]}]
  and the integral agrees with the limit of the integrals
  (and of course on any subset)
  duh: L^1 is weaker than L^infty (on compact set/finite measure)
  (and L^1 is "the right" topology, and obviously guarentees it)
  
  it's -more than strong enough- as a topology

- uniform limit of differentiable is -not- (nec.) differentiable
  (you have essentially -no- (local) control, b/c of oscillations)
  but with the C^1 topology, a C^1 limit is C^1.
  (and again, this is obvious)
  
  it's -not strong enough- as a topology
  
  + -however-, we have some analysis:
    if the derivatives converge uniformly
    (in addition to the function converging),
    then they converge to the right value
      (and indeed the function $f$ is differentiable)

    indeed, by FTC, all we need is:
    - f_n converges at one point
    - f'_n converge in terms of integrals

moral: with The Right Topology on a function space,
  it's complete (what's The Right topology? well, one
  that makes it complete, like duh! Ideally, should be
  "the weakest" topology that makes it complete:
  the discrete metric makes everything complete,
  for instance) -- and usually obviously so.
