Posted by
Russell Standish on
Jul 26, 2007; 2:22am
URL: http://friam.383.s1.nabble.com/DIFFERENTIABILITY-AND-CONTINUITY-tp524276p524280.html
On Wed, Jul 25, 2007 at 12:09:07PM -0600, Nicholas Thompson wrote:
>
>
> Deep down in the tangle of >>>>>'s I just found this gem. The record is
> two confused for me to know who to thank so I will thank you ALL.
>
> > What you have given is the "handwaving" version of the proof. The
> > trouble is that human imagination can easily get us into trouble when
> > dealing with infinities, which is necessarily involved in dealing with
> > the concept of continuity. In the above example, you mention that
> > continuity is important, but say nothing about differentiability. Are
> > you aware that continuous curves that are nowhere differentiable
> > exist? I fact most continuous curves are not differentiable. By most,
> > I mean infinitely more continuous curves are not differentiable than
> > those that are, a concept handled by "sets of measure zero".
>
> OK. I AM BEING CALLED TO A MEAL AND YOU ALL KNOW WHAT HAPPENS WHEN ONE
> DOESNT ANSWER THAT CALL. BAD KARMA
>
> AM I WRONG THAT BOTH CONTINUITY AND DIFFERENTIABILTY OF AT LEAST THE
> primary FUNCTION ARE A PREMISE OF THE MEAN VALUE THEOREM.
Continuity on [a,b] and differentiability on ]a,b[ are the premisses
of the MVT.
>
> MORE TO THE POINT, ARE YOU ALL CONVERGING AROUND THE ASSERTION THAT THE
> MEAN VALUE THEOREM CANNOT BE DONE WITH OUT ALGEBRA? AS OPPOSED THE THE
> VIEW I WAS ENTERTAINING THAT THE MEAN VALUE THEORY IS A LOGICAL PROOF THAT
> IS REPRESENTED ALGEBRAICALLY FOR PEDIGOGICAL PURPOSES.
>
One cannot rigourously deal with the notions of continuity and
differentiability without algebra. Therefore, the verbal version of
MVT is not rigorous, although it works pretty well for an intuitive
understanding. For many people (including physicists, or myself as an
ex-physicist) rigorous understanding is not really needed, we can
trust that mathematicians have done the rigour bit. But the rigorous
expression still needs to be somewhere, and it is probably useful to
have been exposed to mathematical rigour at some point in one's
training.
For pedagogical purposes, I'm not so sure that algebraic
representations are that useful - I much prefer geometric
representations for instance. However, not everyone's thinking style
is the same, and there probably are students that benefit from
algebraic presentation.
This whole discussion started from discussion of a textbook of
analysis for english majors. I'm not all that familiar with teaching
maths to humanities students, but I gather that neither algebraic nor
geometric approaches work with them. For instance, an economist friend
of mine wrote "Debunking Economics", and spelt out all equations in
words. I complained about how much more difficult I found this
presentation, having to mentally translate them back to the original
algebra, and his comment was that I wasn't the target audience. This
was backed up by one of his readers from a humanities background, who
said they found the verbal descriptions much clearer to understand
than if it had been expressed in algebra!
So it is all a question of horses for courses.
> SORRY TO TWIST EVERYBODY'S KNICKERS ABOUT THIS. BUT IRRITATING AS IT MAY
> BE TO YOU ALL, THIS CONVERSATION HAS BEEN VERY HELPFUL TO ME.
>
> NICK
>
> nick
> >
> >
>
>
>
> ============================================================
> 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--
----------------------------------------------------------------------------
A/Prof Russell Standish Phone 0425 253119 (mobile)
Mathematics
UNSW SYDNEY 2052 hpcoder at hpcoders.com.au
Australia
http://www.hpcoders.com.au----------------------------------------------------------------------------