[FOM] A question concerning continuous functions

[sandylemberg@juno.com]

I am not sure exactly what you are getting at but here are some comments.

I opened at random two elementary analysis books to the places indicated
by the words "continuity" or "continuous function" in the index and found
in each what seemed to me an explanation of the kind you asked for. The
books were "Calculus" by Moise and "The Elements of Real Analysis" by
Bartle (original 1st edition). It seems to me that just about every such
book has a similar explanation.

Second, in mathematics, continuity of maps is not usually described in
terms of their graphs. Therefore, in a sense, the 1 dimensional calculus
picture is anomalous with respect to the definition (even though it is
the historical root of the definition). More typical would be a
description of what happens in 2 or more dimensions (Bartle actually uses
this picture). It is this intuition which generalizes to our intuition in
metric and topological spaces.

"A function f is discontinuous at a point a iff there is some fixed
(neighborhood V of f(a)) distance e>0 for which no matter how small a
distance d>0  (neighborhood U  of a) you choose there are always points x
within U which get "tom apart" from a by f and thus sent outside of U." 

The technical definition corresponds perfectly with the intuition.

My last comment has to do with what you said about intuitionism. My
understanding of this is limited, but the idea is that the Dedekind line
is just one of several ways to approach the continuum. The topic of
"infinitesimal analysis" with nilpotent infinitesimals is very much to
the point here. (Poor choice of language, considering the topic.) The
theory has superficial similarity with the invertible infinitesimals of
Robinson but is ontologically more relevant.  This setting involves a
category of smooth objects and smooth maps which are of course
continuous. 

You may know about this, but if you don't, "A Primer of Infinitesimal
Analysis" by John Lane Bell is a good place to start. The continuum he
describes is not composed of points. The law of the excluded middle fails
in cases of locations (or quantities) which are indistinguishably close
but may not be identical. This captures a deep insight into a modern view
of the physical world in which locations and moments in time are
"nonpunctiform" to use Bell's terminology. Continuity of maps is exercise
1.11 on page 25. Even if you don't want to get into it, you should read
his provocative introduction which gives a thumbnail chronology of views
on the continuum and infinitesimals and contains illuminating quotes on
the continuum from the illustrious authors: Aristotle, Leibniz, Kant,
Poincare, Weyl, Brouwer, Thom, and C. S. Pierce.

Sandy