Posted by
Russell Standish on
Jul 26, 2007; 4:02pm
URL: http://friam.383.s1.nabble.com/DIFFERENTIABILITY-AND-CONTINUITY-tp524276p524302.html
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.
--
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