Descriptive set theory

[Cantor's original motivation for his set,
 and for higher cardinals and ordinals etc. etc.]

What are the possible points of continuity of a function?

The answer is:
any $G_\delta$
Pf:
- the points of continuity are a $G_\delta$,
  as they are the intersection of the points where
  the oscillation is $<\epsilon$
  [This shows how oscillation is a good concept:
   it -quantifies- discontinuity]
- conversely, given any closed set,
  you can take the characteristic function of a dense subset,
  and that puts in discontinuity points on the set.
  Add these up (using $2^{-n}$), and you get any $F_\sigma$
  as discontinuities.

What about differentiability etc.?

Cantor came up with functions that failed to be differentiable,
or had discontinuous derivative or whatever,
on the Cantor set
