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Topic: Is zero an approximation?
Replies: 170   Last Post: Dec 8, 2001 12:57 AM

 Messages: [ Previous | Next ]
 Mike Oliver Posts: 1,518 Registered: 12/6/04
Re: Is zero an approximation?
Posted: Dec 5, 2001 6:33 AM

Mike Oliver wrote:

> Not so. The operations +,-,*,/ on R* are, by construction, continuous
> wherever they are defined (for each of them, an open subset of R* x R*;
> not the same open subset for all of them). If you compose them, you
> continue to get partial functions with this property. You can likewise
> extend other functions (say, exp) to have the same property, and again
> this property is closed under composition.
>
> That means that if you see some complicated expression which has
> been built up in this way, you can compute its value in R*, and
> assuming this *gives* you a value (that is, it turns out to be
> defined), then you have correctly computed the limit of the
> expression at that point. Therefore the determination of limits
> is reduced, in many cases, to simple arithmetic -- a substantial
> logical simplification.

I should say here that this will not always be true in the stronger
sense that it could be read -- namely, that if you get a value in
the R* calculation, then the limit working in R will necessarily exist
and be equal to the value computed. If it *does* exist, however, it
certainly must be equal to the computed value.

The strong sense *does* seem to hold in practice; my counterexamples
are rather contrived. E.g. let z : R-->R be the constantly-zero
function, and let s : R-->R be such that s(x) = 1/x^2 (the square
is to finesse the signed-inifinity issue). Now these have unique maximal
continuous extensions to partial functions (though they'll actually
be total in these two cases) from R*-->R*. Working in R*,
we get s(s(z(5))) = 1/(1/0^2)^2 = 1/ oo^2 = 0. However it is
not true in R that lim_{x->5} s(s(z(x))) = 0, because there is
no deleted neighborhood of 5 on which s(z(x)) is defined.

I'm wondering if anyone can come up with some natural criterion, easy
to verify, which guarantees that the strong form holds. I believe it's
enough that the domain of the complicated compound function is open dense
and that that function pulls dense sets back to dense sets; moreover,
this property is preserved under composition. Unfortunately this property
isn't *true* of some of the functions we need in order to represent
some natural expressions as compositions of our basic functions. For
example, the function from R --> R x R given by x |-> <x,x> does
not pull dense sets back to dense sets, and without this function
we can't argue that the function x |-> x^2 is a composition starting
from *, if that makes sense.

Date Subject Author
11/29/01 Jim Bob
11/29/01 Mike Oliver
11/29/01 Charles
11/29/01 Mike Oliver
11/30/01 Ken Oliver
12/1/01 David Libert
11/29/01 Paul Lutus
11/29/01 Jim Bob
11/29/01 Paul Lutus
11/29/01 Mike Oliver
11/29/01 Paul Lutus
11/29/01 Mike Oliver
11/29/01 Paul Lutus
11/29/01 Mike Oliver
11/29/01 Mike Oliver
11/29/01 Paul Lutus
11/29/01 Mike Oliver
11/29/01 Paul Lutus
11/29/01 Mike Oliver
11/30/01 Paul Lutus
11/30/01 Mike Oliver
11/30/01 Paul Lutus
11/30/01 David W. Cantrell
11/30/01 Mike Oliver
11/30/01 Paul Lutus
11/30/01 Mike Oliver
11/30/01 Paul Lutus
11/30/01 Jesse F. Hughes
11/30/01 Paul Lutus
11/30/01 Mike Oliver
11/30/01 Paul Lutus
11/30/01 Daryl McCullough
11/30/01 Mike Oliver
11/30/01 Jesse F. Hughes
11/30/01 The Scarlet Manuka
11/30/01 Paul Lutus
11/30/01 Matt Gutting
11/30/01 Paul Lutus
12/4/01 Matt Gutting
12/4/01 Paul Lutus
12/1/01 David W. Cantrell
12/1/01 Paul Lutus
12/1/01 Mike Oliver
12/2/01 The Scarlet Manuka
12/3/01 Paul Lutus
12/3/01 The Scarlet Manuka
12/3/01 Paul Lutus
12/3/01 The Scarlet Manuka
12/3/01 Paul Lutus
12/3/01 The Scarlet Manuka
12/3/01 Paul Lutus
12/3/01 The Scarlet Manuka
12/3/01 Mike Oliver
11/30/01 Paul Lutus
11/29/01 Mike Oliver
11/29/01 The Scarlet Manuka
11/30/01 Paul Lutus
11/30/01 Jim Bob
11/30/01 The Scarlet Manuka
11/30/01 Mike Oliver
11/30/01 The Scarlet Manuka
11/30/01 Paul Lutus
11/30/01 meron@cars3.uchicago.edu
11/29/01 James Hunter
11/30/01 neoprog
11/30/01 David W. Cantrell
12/1/01 David C. Ullrich
12/2/01 neoprog
12/3/01 David C. Ullrich
11/29/01 Paul Lutus
11/29/01 The Scarlet Manuka
11/30/01 Paul Lutus
11/30/01 The Scarlet Manuka
11/30/01 Paul Lutus
11/30/01 The Scarlet Manuka
11/30/01 Paul Lutus
11/30/01 The Scarlet Manuka
11/30/01 David W. Cantrell
11/30/01 mareg@primrose.csv.warwick.ac.uk
11/30/01 David W. Cantrell
11/30/01 Jon and Mary Frances Miller
11/30/01 Gregory L. Hansen
11/30/01 Doug Norris
11/30/01 James Hunter
11/30/01 Gregory L. Hansen
11/30/01 James Hunter
11/30/01 Virgil
11/30/01 Paul Lutus
11/30/01 James Hunter
11/30/01 Paul Lutus
11/30/01 James Hunter
11/30/01 Paul Lutus
11/30/01 mareg@primrose.csv.warwick.ac.uk
12/1/01 Jim Bob
12/2/01 Tubby Chubbs
12/2/01 Avoid normal situations.
11/30/01 neoprog
12/5/01 Randy Poe
11/30/01 Julian Fondren
12/2/01 The Scarlet Manuka
11/30/01 Julian Fondren
11/30/01 Paul Lutus
11/30/01 Jim Bob
11/30/01 Paul Lutus
11/30/01 Jim Bob
11/30/01 Paul Lutus
12/1/01 Christian Ohn
12/1/01 Charles Dillingham
12/1/01 Mike Oliver
12/2/01 Charles Dillingham
12/2/01 Mike Oliver
12/2/01 Mike Oliver
12/2/01 Mike Oliver
12/2/01 Bob Kolker
12/2/01 Mike Oliver
12/2/01 Bob Kolker
12/2/01 Mike Oliver
12/2/01 Bob Kolker
12/2/01 Mike Oliver
12/3/01 David W. Cantrell
12/3/01 Bob Kolker
12/3/01 The Scarlet Manuka
12/3/01 Bob Kolker
12/3/01 The Scarlet Manuka
12/3/01 meron@cars3.uchicago.edu
12/2/01 Bob Kolker
12/2/01 Mike Oliver
12/3/01 Bob Kolker
12/3/01 The Scarlet Manuka
12/3/01 Mike Oliver
12/5/01 Mike Oliver
12/7/01 David W. Cantrell
12/8/01 Mike Oliver
12/3/01 Charles Dillingham
12/3/01 Mike Oliver
12/5/01 Charles Dillingham
12/5/01 Mike Oliver
12/5/01 Mike Oliver
11/29/01 Bryan Reed
11/30/01 Jim Bob
11/30/01 Paul Lutus
11/29/01 Uncle Al
11/29/01 Stephen Norris
12/2/01 James Hunter
11/30/01 glenn
11/30/01 James Hunter
11/30/01 Rushtown
11/30/01 Jesse F. Hughes
11/30/01 Virgil
11/30/01 Bob Kolker
11/30/01 Julian Fondren
11/30/01 Paul Cardinale
11/30/01 Paul Lutus
11/30/01 franz heymann
11/30/01 Paul Lutus
12/3/01 Paul Cardinale
12/3/01 Paul Lutus
12/5/01 Paul Cardinale
12/5/01 Paul Lutus
12/6/01 Paul Cardinale
12/6/01 Paul Lutus
12/7/01 Paul Cardinale
12/7/01 Paul Lutus
11/30/01 Stephen Norris
11/30/01 David W. Cantrell
12/1/01 mensanator
12/3/01 Juuichirou Tendou
12/6/01 Keith Ramsay