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DIFFERENTIABILITY AND CONTINUITY

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Nick Thompson

DIFFERENTIABILITY AND CONTINUITY

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.

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.

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
>
>

Russell Standish

DIFFERENTIABILITY AND CONTINUITY

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
----------------------------------------------------------------------------

Phil Henshaw-2

DIFFERENTIABILITY AND CONTINUITY

In reply to this post by Nick Thompson

Nick,
There might be several definitions of continuity, that correspond to
different properties, some included in each other and some not. My guess
is that the non-differentiable type being referred to, but not named or
described, is different from the differentiable one(s) that one more
commonly runs into, and given the complicated ways people can define things
maybe there are several kind of choices for guessing what's being talked
about. The one mentioned is not defined it seems, except by way of asking
the poor reader for a "gee whiz oh gosh" response of some sort. ...so
belaboring the point... is there something missing??

On 7/25/07, Nicholas Thompson <nickthompson at earthlink.net> 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.
>
> 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.
>
> 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
>
>

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Frank Wimberly

DIFFERENTIABILITY AND CONTINUITY

See for example:

http://www.math.tamu.edu/~tvogel/gallery/node7.html

---
Frank C. Wimberly
140 Calle Ojo Feliz??????????????(505) 995-8715 or (505) 670-9918 (cell)
Santa Fe, NM 87505???????????wimberly3 at earthlink.net
-----Original Message-----
From: [hidden email] [mailto:[hidden email]] On
Behalf Of Phil Henshaw
Sent: Thursday, July 26, 2007 1:58 PM
To: nickthompson at earthlink.net; The Friday Morning Applied Complexity
Coffee Group
Subject: Re: [FRIAM] DIFFERENTIABILITY AND CONTINUITY

Nick,
There might be several definitions of continuity, that correspond to
different properties, some included in each other and some not.????My
guess is that the non-differentiable type being referred to, but not
named or described, is different from the differentiable one(s) that one
more commonly runs into, and given the complicated ways people can
define things maybe there are several kind of choices for guessing
what's being talked about.?? The one mentioned is not defined it seems,
except by way of asking the poor reader for a "gee whiz oh gosh"
response of some sort.??? ...so belaboring the point... is there
something missing??

?
On 7/25/07, Nicholas Thompson <nickthompson at earthlink.net> 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.

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.

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

Phil Henshaw-2

DIFFERENTIABILITY AND CONTINUITY

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. 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). Another is the class of curves formed by having a rule
for finding point betweem any two, but having no formula. The latter
is interesting because it's everywhere discontinuous, but fairly easy to
make differentiable... :-)

Phil Henshaw ????.?? ? `?.????
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
680 Ft. Washington Ave
NY NY 10040
tel: 212-795-4844
e-mail: pfh at synapse9.com
explorations: www.synapse9.com

> -----Original Message-----
> From: friam-bounces at redfish.com
> [mailto:friam-bounces at redfish.com] On Behalf Of Frank Wimberly
> Sent: Thursday, July 26, 2007 4:59 PM
> To: 'The Friday Morning Applied Complexity Coffee Group'
> Subject: Re: [FRIAM] DIFFERENTIABILITY AND CONTINUITY
>
>
> See for example:
>
> http://www.math.tamu.edu/~tvogel/gallery/node7.html
>
>
> ---
> Frank C. Wimberly
> 140 Calle Ojo Feliz??????????????(505) 995-8715 or (505)
> 670-9918 (cell) Santa Fe, NM 87505???????????

wimberly3 at earthlink.net -----Original > Message-----

> From: friam-bounces at redfish.com
> [mailto:friam-bounces at redfish.com] On Behalf Of Phil Henshaw
> Sent: Thursday, July 26, 2007 1:58 PM
> To: nickthompson at earthlink.net; The Friday Morning Applied
> Complexity Coffee Group
> Subject: Re: [FRIAM] DIFFERENTIABILITY AND CONTINUITY
>
> Nick,
> There might be several definitions of continuity, that
> correspond to different properties, some included in each
> other and some not.????My guess is that the
> non-differentiable type being referred to, but not named or
> described, is different from the differentiable one(s) that
> one more commonly runs into, and given the complicated ways
> people can define things maybe there are several kind of
> choices for guessing what's being talked about.?? The one
> mentioned is not defined it seems, except by way of asking
> the poor reader for a "gee whiz oh gosh" response of some
> sort.??? ...so belaboring the point... is there something missing??
>
>
> ?
> On 7/25/07, Nicholas Thompson <nickthompson at earthlink.net> 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.
>
> 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.
>
> 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
>
>
>
> ============================================================
> 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
>
>

Russell Standish

DIFFERENTIABILITY AND CONTINUITY

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 hpcoder at hpcoders.com.au
Australia http://www.hpcoders.com.au
----------------------------------------------------------------------------

Frank Wimberly

DIFFERENTIABILITY AND CONTINUITY

In reply to this post by Phil Henshaw-2

Well, according to any conventional definitions differentiability
implies continuity.

---
Frank C. Wimberly
140 Calle Ojo Feliz (505) 995-8715 or (505) 670-9918 (cell)
Santa Fe, NM 87505 wimberly3 at earthlink.net

-----Original Message-----
From: [hidden email] [mailto:[hidden email]] On
Behalf Of Phil Henshaw
Sent: Thursday, July 26, 2007 8:29 PM
To: 'The Friday Morning Applied Complexity Coffee Group'
Subject: Re: [FRIAM] DIFFERENTIABILITY AND CONTINUITY

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. 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). Another is the class of curves formed by having a rule
for finding point betweem any two, but having no formula. The latter
is interesting because it's everywhere discontinuous, but fairly easy to
make differentiable... :-)

Phil Henshaw ????.?? ? `?.????
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
680 Ft. Washington Ave
NY NY 10040
tel: 212-795-4844
e-mail: pfh at synapse9.com
explorations: www.synapse9.com

wimberly3 at earthlink.net -----Original > Message-----

============================================================
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