More -key- calc things
-------------------

- global/local/infintesimal ***
  this is a -key- organizing principle in calc;
  it's -the- key one w/r/t to geometry
  [standard EG is: increasing implies all secants
   positive slope, so all tangents -semi-positive slope
   (see how closure arises naturally);
   conversely, all tangents (or almost all, in sense of measure)
   positive implies increasing,
   and all semipositive implies semi-increasing;
   global -> local -> inf at first (and easy),
   inf -> global second, using calc]
-------------------
- functions
  ***
  The -difference- between a function and its graph;
  not -identifying- functions and their graph.
  [a key technique is to give functions that don't have
   useful graphs, like char. func. of rationals]

  This is a good topic, b/c it's related to an old question:
  what is a function?
  We distinguish between a -function- and a -path-.
    (a path is just -2- functions)

  In particular, a discussion of tangent lines to paths explains
  why $\sqrt[3]{x}$ has a ``vertical tangent'' at 0:
  qua -function-, it's simply -not differentiable-
    (think of it as a mapping, for instance)
  qua -path-, the -image- is differentiable,
    and given a good -parametrization- (like $(y^3,y)$)
    it is differentiable, but a bad parametrization
    (like $(x,\sqrt[3]{x})$) is not differentiable:
    and indeed we can give non-differentiable parametrization
    of the real line (like $\sqrt[3]{x}$)

-------------------
- limits
  + notion of limit in punctured neighborhood
    (and oscillation, my favorite omitted limit topic!
     including lim sup and lim inf! I -swear-!)
    ...and that this is exactly what we need for -tangents-:
    the ``slope of secant line'' makes perfect sense -away- from
    the point in question: we want to know if we can extend it
    continuously to that point.
  
  + punctured vs. unpunctured
    [this is a -hair-splitting- distinction,
     but that makes it all the more confusing when it -isn't- made;
     students REALLY benefit from hairsplitting distinctions:
     if you don't make them, they get really confused and conflate
     different things]
    for -continuity-, it's better to work with unpunctured
    neighborhoods: a function is continuous iff it has an
    -unpunctured- limit there
    [which, yes, is equivalent to having a punctured limit
     and the punctured limit equally the value at the point;
     but see how this is extra steps?]

    for -differentiability-, you have to work with punctured limits;
    
    one of the reason calc books are unclear on continuity is b/c
    they don't really care about it: their real goal is
    differentiability, which is why they work always with punctured
    limits

    [formally, we have two directed systems:
     the punctured neighborhoods of a
     and the unpunctured ones;
     we're taking the limit over these...]

    ...and note that since you can't plug infinity into functions
    (until you do complex analysis ;-), limits to infinity
    are always punctured ones, which is why they're in many ways
    -easier- to understand.
  
  + (individual, finitely many) points don't matter
    Intuitively, we know that if two functions agree at cofinitely
    many points, all their limits are the same.
    This should (eventually) be proven, preferably by the students
    doing induction and using the below analysis.
    The two key props are:
    - [other points don't matter]
      if f and g agree except at b,
      then their limit at $a\neq b$ is the same
      (...and they simultaneously converge/diverge)
      [more precisely, if they agree on a neighborhood of $a$,
       then their limit at $a$ is the same]
    - if $f$ and $g$ agree except at $a$
    ...and these should be stated informally at first, then later proven.
    They can be summarized as:
    if $f$ and $g$ agree on a (small enough) punctured neighborhood of $a$,
    then their limit as you approach $a$ is the same.
    ...and this is obvious (and also easy to prove): limits
    just depend on value on a small enough punctured neighborhood.
    
-------------------
- algebra
  + limits
    being able to do easy/stupid computations of limits
    roughly:
    - if continuous, just plug in numbers
    - if continuous except that point but removable discontinuity,
      (eg, x/x), just plug in (to the continuous version)
    - in particular, if rational, just plug in and/or cancel
    - rationals at infinity: look at dominant term
    - L'Hopital's rule
      [later of course a cute proof]
      + ...including variants, like $0^0$, $1^\infty$, etc.
        (yeah, useful for example of $e$;
         also give EG of Chris Kringle problem,
         to see that this is really useful (for suitable
         notions of "useful"))
    - ad hoc techniques, especially:
      + boundedness
        (lim of (x+sin x)/x = 1
        [also useful for doing/proving rationals at infinity]

  + also, give EGs to note that in general, limits are very hard to
    compute! (esp. of infinite sums...)
    it's only when we have a nice structure/geometry/symmetry
    that we can actually do computations or understand what's going
    on. (give EG of euler's constant, for instance, and tell my
    little story -- which helps motivate rationality and limits)
    (and of course the classic question:
     is the limit of rational numbers a rational number?
     no, of course not)
