Re: terminology in definitions of limits

[Michael Barr <barr@math.mcgill.ca>]

This is getting peripheral to the main point.  AS far as I recall, I
thought of T as a test object.  As for epsilon-delta, Bishop required that
delta be prescribed as a constructible function of epsilon in order that a
function be continuous.  He required that the convergence be uniform on
every closed interval, so that this function on a closed interval was
independent of the points in the interval.

Michael

On Wed, 21 Jan 2009, Charles Wells wrote:

> Calculus teachers do something similar when they make an epsilon-delta proof
> into a game:  The opponent picks an epsilon (the test object) and you have
> to come up with a delta.
> There is one big difference between epsilon-delta proofs and limits.  To
> show that something is a limit you have to find, for each test object, the
> unique arrow specified by the definition of limit.  Thus you are producing a
> function (indeed, a bijection).   The delta for a given epsilon is not unique,
> and so there is no natural function giving a delta for each epsilon.  I am
> pretty sure this makes epsilon-delta proofs harder for non-talented students
> than proving something is a limit.  I know some calculus teachers talk about
> there being a function that takes epsilon to delta, but I suspect it is a
> mistake to bring that up.
>
> Charles Wells
>