\section{Definition of continuity}

The definition of continuous function \emph{should} strike you as weird and
unintuitive at first:
we define continuity at a point, and then say that a function is continuous if
it's continuous at each point.

The reason we do it this way is so that continuity is an \emph{infintesimal
property} of a function
...
(but then functions that are globally continous by this definition are -weird-
this took people a while to figure out;
Peano curve
sawtooths and nowhere differentiable functions
function spaces and ``generic'' functions

