TODO

* Make everything be in some subsection, the "introduction" divisions in
  pretext may be screwing up and just dropping certain things on the floor.
  This requires a new edition since there are going to be probably page
  number changes.  If this is done, we could drop the manual introduction
  %mbxINTROSUBSECTION thing and do it automatically again perhaps.

HTML/PreTeXt:

* Check the equation with the aligned in the Implicit function theorem
  section in volume 2 if it works with the PreTeXt conversion

* Check everything.

* https://pretextbook.org/doc/guide/html/processing-directory-management.html?

* Make sure to use mathtools in the PreTeXt?

MAYBEs (possibly next, possibly other edition, possibly never):

* Vol II: Does Lemma 10.3.9 really need to be done for closed balls at all?
  Seems like it just confuses things a bit.  Need to check if that is used
  in any of the exercises or later, but it's not referred to.
  The only reason for having used closed balls was the proof which was
  not quite right anyway, no it proves open first.

* Vol I: In 7.6, add an exercise for Picard with parameters, either
  continuous or C^1 perhaps that could use 6.2, but this kind of
  requires partial derivatives just to state.

* Vol I: Add another example for computing limits using the theorems to
  subsection 2.2.2?  Perhaps not needed, but might be good.
  (thx Manuele Santoprete)

* Vol I: Perhaps be always a bit more explicit with the domain on both
  "uniformly continuous" and "Lipschitz"
  (thx Manuele Santoprete)

* Vol I: A bit more about Dedekind vs Cauchy completeness?
  (thx Manuele Santoprete)

* Vol I/II: Add more names to the theorems/propositions.  E.g.
  From Manuele Santoprete:
  "I think it is good to give names to theorems, since it make it easier to
   communicate math. There are several spots where one can give a name to the
   theorems. The most important one is Proposition 2.1.10. This is often called
   the "Monotone Convergence Theorem". I think at least this name should be
   used.  Also the min-max theorem is often called extreme value theorem. I
   think it makes sense to mention this, since many students are familiar with
   the term from calculus.  Bartle's book has some very nice names. Another one
   is the Cauchy Convergence Criterion (Theorem 2.4.5). There are others as
   well, but I did not make a note of them. Moreover,  Lemma 2.2.3 sometimes is
   called the Comparison Theorem."
  MCT is done, but more should be done

* Vol I: Perhaps avoid the whole "well ordering of N" and "induction"
  equivalence bit.  We could just assume induction as axiom and have well
  ordering as a proposition.  Might require replacing the exercise 0.3.18
  as we might want to just prove this in text.

* Vol II: Add the "crash course on Lebesgue" chapter

* Vol II: Add a chapter on submanifolds and differential forms.

* Add appendix to Vol I on basic logic, peano axioms, etc.

* Add appendix to Vol I on construction of the reals.  Perhaps start
  with construction of $\Q$ and then finish with Dedekind cuts.
  This could also just be section 1.6?  Note that this conflicts a bit with
  exercise 1.2.15 if we use Dedekind cuts.

* Vol I: Possibly change definition of interior to be the more
  topological exercise 7.2.14, then would need the current definition to be
  a proposition.  This would make things more uniform in presentation with
  closure, but a number of exercises would change.
  The point of the current definition is that it mirrors the definition
  of open set, so it is nice for other reasons.  I can see pros and cons,
  it seems not worth it.

* Vol I:
Suggestion from Arthur Busch for theorem 3.4.4:
"
Note that |x_n-y_n| < 1/n allows you to use the squeeze lemma on {x_n-y_n}, and conclude this sequence converges to 0.  Then apply B-W to {y_n} to get a subsequence {y_{n_k}} converging to some c in the domain.  Since the subsequence {x_{n_k} - y_{n_k}} must converge to 0, you can use the algebraic properties of limits to conclude that {x_{n_k}} also converges to c.  
You can then finish the proof with a quick case argument:  case 1:  Either {f(x_{n_k})} diverges or {f(y_{n_k})} diverges, which immediately makes f discontinuous at c.  case 2:  both these sequences converge, and then {f(x_{n_k}) - f(y_{n_k})} converges to Lx - Ly ≥ ε > 0 so at least one of these limits is different than f(c), making f discontinuous at c.

* Vol I:
Suggestion from Jacob Bernstein:
"
One thing that Strichartz did that I liked (and wasn't in your book as far as I could tell) was prove that if a function had positive derivative at x_0, then it is "locally monotone at x_0" in the sense that f(y)<f(x_0)<f(x) for x_0+\epsilon>x>x_0>y>x_0-\epsilon.  He compares this with the stronger result you get from the MVT.
"

* Mark optional sections / subsections better.  Perhaps "*" in subsection
  title, but then make sure it is really not required elsewhere.

* Maybe mark examples that are used in the sequel somehow. (This may be a
  daunting task, and more imprecise than one imagines, as what does "use"
  mean: What if it is used in a later example that isn't used otherwise?
  What if it is used in an optional section?  What if it appears in an
  offhand remark?)

* Maybe mark exercises whose proof is used in the text later.  Same issues
  as above.

* Maybe: Vol II: Add full proof of Green's.  It is possible that it is
  just making things too difficult.  Really in most cases, the version given
  can be used.
  Since P and Q are defined in a larger set is to approximate U by a
  polygon, or lots of small squares.  For this we need an approximation for
  the path integrals.  Once P and Q are close to constant (uniformly
  continuous) in any small neighborhood, then locally we can use
  independence of path.  So use little rectangles coming from the bit where
  the boundary is of measure zero.  Putting the whole thing together still
  seems a bit tricky to do rigorously, especially if the path has a corner.
  Idea: Perhaps add a remark about an idea of the proof?