On Thu, Jul 26, 2007 at 10:28:59PM -0400, Phil Henshaw wrote: > Yes, I thought that might be the type, though I think there are also > others. Allowing 'continuous' as a general term to include curves that > freely include discontinuities of direction redefines the term in most > people's minds, and is the reason for the 'surprise' that it doesn't > have the usual properties.
Really? I think most "naive" people (ie those without any exposure to differential calculus) would say that a bent stick (or the hip roof for that matter) are continuous. People would only think that differentiability was always associated with continuity if they had a bit of calculus exposure, but not enough to see that continuity is insufficient for differentiability. Ther's a couple other interesting classes of > continuities worth exploring, the impose constraints on the, as well as > make the analysis complicated. One is the class of continuous curves > that are consistent with energy conservation (they can't have infinite > derivatives). A curve with an infinite derivative at a point is not differentiable at that point (by definition). > Another is the class of curves formed by having a rule > for finding point betweem any two, but having no formula. I'm curious to know what you mean by this. Do you have an example? > The latter > is interesting because it's everywhere discontinuous, but fairly easy to > make differentiable... :-) > I would be surprised, as differentiability requires continuity. Perhaps you are thinking of some general notion of derivative, like Nottale's fractal spacetime stuff. However, that is anything but easy. -- ---------------------------------------------------------------------------- A/Prof Russell Standish Phone 0425 253119 (mobile) Mathematics UNSW SYDNEY 2052 [EMAIL PROTECTED] Australia http://www.hpcoders.com.au ---------------------------------------------------------------------------- ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org
