\chapter{Continuity}
continuity is both 0th order differentiable
(hence the notation C^0, C^1, etc.)
...and more general
(hence the notation C)
defns of continuity
many many;
for R -> R,
order and metric both work,
and sometimes one or the other is easier
"smeared out"
= intersection of closure of images
\subsection{Monotone functions and continuity}
1/x is continuous;
the general proof is: "invertible monotone -> continuous";
only way fail to be continuous is b/c of jumps
devil's staircase,
and (any section thereof is monotone discontinuous)]
[cool: a sequence converges iff oscillation = 0; lim inf = lim sup (and
both exist);
a function is continuous at x iff osc = 0;
recall that osc is good b/c rather than having 2 quantifiers,
you've packed it into a concept and a limit]
[a sequence is constant iff min = max (& both exist; NB: also define/say
bounded)]
abstraction:
pointless topology and locales
What's the point? (of pointless topology)
abstracts away from individual points, duh
--------------------------------------
Remember POVs on topology:
- convergence (of sequences, of nets):
this is all about points & analysis
- open sets
--> locales, sheaves, etc.
this is very pointless & algebraic