As noted elsewhere (I think) there are vying perspectives in calc.
That is, these are serious -underlying- issues:
if you ignore them, they'll bite you or confuse you.

One question is how to present the topology:
- as order or as metric?
  (accordingly as Dedekind/Cauchy construction,
   or using order/metric balls)

Another is what functions do you want to prove stuff about?
- $C^1$
  rather geometric, can use the FTC freely;
  generally assume uniform convergence

- everywhere differentiable (or a.e.!) w/ intergrable derivative
  can use the FTC if you're careful, and only need $L^1$
  convergence
  
- everywhere differentiable
  here you need to use MVT, and often need to assume
  uniform convergence, not just $L^1$
  Rudin uses this, of course.
  It's harder and trickier, IMHO.

--------------------------------
Of the last 2, neither is more general than the other.
However, I prefer the former, using FTC:
that is, if you have a very good integral, then any function
that's everywhere differentiable is integrable,
and you have the FTC in full generality:
it's morally why everything is happening.
--------------------------------
These choices show up several places:
- derivatives of limits
- L'Hopital's rule
  (I like by FTC; others use Cauchy MVT, which is a little opaque...)
--------------------------------
Another big and deep question is:
- what do we mean by "integration"?

That is, on the real line, there's two notions:
- lebesgue
  That is, we consider the space as -only- a measure space.
- intervals
  Here we also keep track of intervals.

This makes a difference:
(sin x)/x is -not- lebesgue integrable, but it is integrable if you
keep track of intervals.

Likewise, every derivative is interval-integrable (virtually
tautologically), but needn't be lebesgue integrable.

Similarly, we have a notion of "conditionally summable sequences"
vs. absolutely summable.

This problem/issue underlies a -lot-:
- conditional vs. absolute summability
- improper integrals
- many of the technical contortions/details
  in the Lebesgue integral: every lebesgue integrable function
  is -absolutely- integrable
  (and indeed this essentially characterizes them among all
   "generally integrable" functions)
[note also that the domain being infinite vs. the function taking
 infinite values are different issues;
 the function taking infinite values isn't too bad if you're
 talking about absolute integrability, but otherwise...oy!]
[there's also the tricky notion of "principal value", EG:
 $\int_0^\infty \log x = 0$, in some sense.]

Q: how do these "non-absolute integrals" work for $L^2$ and such?

At least at the intro calc level, I think:
- discussing that summing series is just improper integration
  over the natural numbers
- showing rearrangement
- saying that b/c of rearrangement, in -some- sense
  non-absolute sums don't make sense
  ...but OTOH, they clearly -do- make sense, in a delicate way.
- showing (sin x)/x, and how it's a function that's like
  an alternating series (and clearly has an improper integral)
- [maybe: saying that every derivative is integrable in some sense,
   and the FTC holds in large generality]
...is quite enough.
----------------------------------------------------------------------
metric vs topology vs measure

There's many different structures on the reals, and it's valuable to
keep them all in mind, and how they interact.

The metric structure is familiar, esp. due to endless drill:
it's the distance between points, and the infintesimal metric.

The metric yields a topology by metric topology (from which we get the
epsilons and deltas), and a (Borel) measure:
hence if you preserve the metric, you preserve the topology and
measure.

Topology and measure are two weakening (two -key- weakenings) and
generalizations of notions like metric:
you can preserve topology but not measure,
or other way around.
[amusingly, on the real line, if you preserve topology and measure,
 you're an isometry (preserving measure implies jacobian is $\pm 1$,
 which on the real line means you're differentiable and
 in $\Aff(O(1))$, IE $ax+b$ where $a=\pm 1$).
 This isn't true in more dimensions: take a skew in 2D
 (you can even be non-differentiable: skew each slice differently)]

preserving topology is ``continuous'' -- but squishy, like rubber
sheets
preserving measure is like an incompressible fluid

[measure is what you care about when doing integration,
 which is why it matters so]

EGs to illustrate:
$1/x$ is continuous but not measure-preserving (on $x>0$):
  from POV of topology, $0$ and $\infty$ are the same kinda infinity:
  there's no difference between infinitely far away and a hole
  [for manifolds there is a notion of ``wild ends'', and some kinda
   difference, but that's much more intricate]
  OTOH, from POV of measure, they're very different
