[FOM] A question concerning continuous functions

[Alex Simpson]

Dana Scott writes:

> In other words, no jumps, no lifting the chalk (pen/pencil).  As you
> draw, you have to stay close to previous points.  Is there anything
> deeper here?  Does the definition not make an intuitive idea rigorous?

One property that continuous functions in Arnon Avron's sense 
intuitively enjoy is that the graph of any continuous function
should, on a closed interval, have a well-defined finite "length 
along the curve" - at least assuming that we ought to be able to
draw the graph with bounded chalk velocity. So "no jumps" is
not necessarily the whole story here.

On the other hand, there are presumably many alternative
consistent stories possible. As Sandy Lemberg argues, having a 
notion of continuity that applies just as well at higher 
dimensions is fundamental. This would seem a very good
test for any alternative proposal for defining continuity.

Alex Simpson

Alex Simpson, LFCS, Division of Informatics, Univ. of Edinburgh
Email: Alex.Simpson at ed.ac.uk           Tel: +44 (0)131 650 5113
Web: http://www.dcs.ed.ac.uk/home/als  Fax: +44 (0)131 667 7209