Date: Nov 11, 2012 12:57 PM
Author: Joe Niederberger
Subject: Re: How teaching factors rather than multiplicand & multiplier<br> confuses kids!

Robert Hansen says:
>There is no such thing as a "common sense view of continuity".

This didn't take long to find via Google:
http://www.maa.org/pubs/Calc_articles/ma004.pdf

In that paper we see Leibniz coming very close to the modern concept, but still no cigar. How could that be if he had no understanding of continuity *prior* to the modern formulation? Well, of course he had an intuition, that was based on something entirely other than the modern formulation.

Here's another paper that uses the pencil idea to illustrate the inutitive notion:
http://www.math.harvard.edu/~nasko/documents/topology_and_continuity.pdf

Note the author quickly points out that the formal definition captures cases that are completely outside the purview of the intuitive notion: "functions that do not jump at an isolated points as above, but infinitely often in a dense manner". Completely unintuitive and non-commen-sensical.

That the views two lead to different phenomena only reinforces the fact that the common sense view and the formal view are not at all the same animal.

Joe N