[FOM] A question concerning continuous functions

[sandylemberg@juno.com]

"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" -"

This is why we have the notion of bounded variation. Real analysis is
replete with counterintuitive or anomalous examples. In a very real
sense, most examples fall into this category! The theory of the continuum
and foundations of real analysis is, for me, frankly a deep and
mysterious subject.

Sandy

On Fri, 31 Jan 2003 05:45:08 +0000 (GMT) Alex Simpson <als at inf.ed.ac.uk>
writes:
> 
> 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  
> 
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