[FOM] A question concerning continuous functions

[John Goodrick]

On Mon, 27 Jan 2003, Arnon Avron wrote:
> 3) Returning to the comparison with Church Thesis, there is one
>    big difference in that (at least in my eyes) the intuitive definition
>    of a continuous function  is *not* vague or imprecise  (certainly not
>    as vague and imprecise as the notion of a "mechanically computable
>    function" had been before Church Thesis was formulated). It is
>    completely sufficient, in fact, for conveying and understanding the
>    notion we have in mind.

Really?  I've always found the "chalkboard-drawing" definition of
continuity way too vague.  For instance, how would you use that definition
to decide if the following functions are continuous?

1. A continuous, nowhere-differentiable function.

2. f(x) = x^2 sin(1/x), if x is nozero, and f(0) = 0.

These are both continuous by the analytic definition, but most people
would probably say they couldn't draw them on a blackboard... but then,
what does it mean to "draw" any function?

Most people's notion of a function whose graph you could draw, without
lifting the chalk, probably corresponds to something a little stronger
than continuity, like maybe piecewise continuously differentiable and
having a bounded first derivative (where this is defined).

-John