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Topic: Cantor's first proof in DETAILS
Replies: 85   Last Post: Dec 10, 2012 7:23 AM

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ross.finlayson@gmail.com

Posts: 1,186
Registered: 2/15/09
Re: Cantor's first proof in DETAILS
Posted: Dec 9, 2012 1:53 PM
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On Dec 8, 10:35 pm, Virgil <vir...@ligriv.com> wrote:
> In article
> <2e2a7c1e-973c-4b07-b8f1-cce24261a...@jl13g2000pbb.googlegroups.com>,
>  "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote:
>
>
>
>
>
>
>
>
>

> > On Dec 8, 9:56 pm, Virgil <vir...@ligriv.com> wrote:
> > > In article
> > > <893b9130-50b0-4ffe-aff1-313d15bfc...@r10g2000pbd.googlegroups.com>,
> > > "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote:

>
> > > > On Dec 6, 9:24 pm, Virgil <vir...@ligriv.com> wrote:
> > > > > In article
> > > > > <97829085-9b08-479c-b693-fde704b4f...@nl3g2000pbc.googlegroups.com>,
> > > > > "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote:

>
> > > > > > On Dec 5, 9:05 pm, Virgil <vir...@ligriv.com> wrote:
> > > > > > > In article
> > > > > > > <5312c40d-7490-4838-b49c-573a9f2e1...@i2g2000pbi.googlegroups.com>,
> > > > > > > "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote:

>
> > > > > > > > On Dec 4, 1:15 pm, Virgil <vir...@ligriv.com> wrote:
> > > > > > > > > In article
> > > > > > > > > <42cabcca-089d-456f-837a-c1d789bda...@jj5g2000pbc.googlegroups.c
> > > > > > > > > om>,
> > > > > > > > > "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote:

>
> > > > > > > > > > And Heaviside's step is continuous,
> > > > > > > > > > now. For that matter it's a real function.

>
> > > > > > > > > I already said that the step function is a real function, I
> > > > > > > > > only
> > > > > > > > > objected to your claim that it was a continuous function.
> > > > > > > > > --

>
> > > > > > > > Heh, then you said it wasn't, quite vociferously
>
> > > > > > > I objected to it being called continuous. possibly vociferously,
> > > > > > > but
> > > > > > > your claim that it was continuous deserved vociferous objection.

>
> > > > > > > : you were wrong
> > > > > > > Don't you wish!

>
> > > > > > > , and
>
> > > > > > > > within the course of a few posts wrote totally opposite things.
> > > > > > > > Your
> > > > > > > > memory fails and that's generous, not to mention you appear
> > > > > > > > unable to
> > > > > > > > read three posts back.

>
> > > > > > > > And everybody sees that.
>
> > > > > > > > Then as noted Heaviside's step, a real function, can be simply
> > > > > > > > drawn
> > > > > > > > classically: without lifting the pencil.

>
> > > > > > > Not outside of Rossiana.
>
> > > > > > >http://en.wikipedia.org/wiki/Heaviside_step_function
> > > > > > > The Heaviside step function, or the unit step function, usually
> > > > > > > denoted
> > > > > > > by H (but sometimes u or ?), is a discontinuous function whose
> > > > > > > value is
> > > > > > > zero for negative argument and one for positive argument. It seldom
> > > > > > > matters what value is used for H(0), since H is mostly used as a
> > > > > > > distribution.

>
> > > > > > > It's continuous that way.
>
> > > > > > > Not according to Wiki, whom EVERONE here, except possibly WM,
> > > > > > > trusts far
> > > > > > > more than they trust Ross.

>
> > > > > > > See that phase "discontinuous function"?
>
> > > > > > > Or maybe your as blind as you are thick.
> > > > > > > --

>
> > > > > >http://en.wikipedia.org/wiki/Heaviside_step_function
>
> > > > > From wiki:
> > > > > The Heaviside step function, or the unit step function, usually denoted
> > > > > by H (but sometimes u or ?), is a DISCONTINUOUS function whose value is
> > > > > zero for negative argument and one for positive argument. It seldom
> > > > > matters what value is used for H(0), since H is mostly used as a
> > > > > distribution. Some common choices can be seen below.

>
> > > > > > * ''H''(0) can take the values zero through one as a removal of the
> > > > > > point discontinuity, preserving and connecting the neighborhoods of
> > > > > > the limits from the right and left, and preserving rotational
> > > > > > symmetry
> > > > > > about (0, ).

>
> > > > > Except that the value of the Heaviside step function AT zero cannot be
> > > > > chosen so as to make its limit as x increases towards zero though
> > > > > negative values become equal to the limit as x decreases through
> > > > > positive values towards zero, which would be necessary to make the
> > > > > function continuous at zero according to every standard definition of
> > > > > continuity.

>
> > > > > One wonders whether Ross knows what continuity reall is all about.
>
> > > > > >http://en.wikipedia.org/wiki/Oliver_Heaviside
>
> > > > > > Looks good to me.
>
> > > > > Try getting your eyes tested, and if that doesn't clear things up, get
> > > > > your brain tested.
> > > > > --

>
> > > > You describe a particularly strong condition of continuity, there are
> > > > weaker ones, that leave the classical notion that if you can draw it
> > > > in one non-crossing stroke it's continuous. Heaviside's step,
> > > > connected, is in a sense continuous.

>
> > > Then what is your example of a discontinuity?
>
> > > My "strict" one is the only level of "condition of continuity" that I
> > > have found in any elementary calculus text that I have, or in any
> > > advanced text, for that matter. For example, none of the several texts
> > > on calculus written by Tom Apostol, accept anything less than the sort
> > > of definition I would require:
> > > for a real function, f, defined on an open set containing 0
> > > to be continuous at zero, it is necessary that
> > > (1) the function must have a value, f(0), at zero.
> > > (2) the function must have a finite limit as x increases to 0.
> > > (2) the function must have a finite limit as x decreases to 0.
> > > (4) both limits must equal f(0).

>
> > > I Googled for "continuity at a point" and found over 20 million sites.
> > > Those I sampled all agreed with me And Apostol, and I dare sat that none
> > > of them disagree to the point of calling any function which has value
> > > zero at every negative argument and value one at every positive argument
> > > can be anything but definitely DIScontinuous at zero.

>
> > > It you ever find one of those 20 million sites, not written by
> > > yourself,which says otherwise, please post its URL in refutation.
> > > Otherwise stop lying!
> > > --

>
> > Euler: and he was blind.
>
> >http://en.wikipedia.org/wiki/Leonhard_Euler
>
> I have looked at your reference to Euler and found nothing in it that in
> any way contradicts the definition of continuity I expressed above.
>
> So that, since I am not blind, Ross must be, to be able to se what is
> not there.
>


What you missed was that I agreed that, given the definition of
"continuous" as being right- and left-continuous at each point and
with the same limit, that H-connected is "naturally continuous". I
don't claim that function has properties it doesn't. It's the straw-
man.

http://en.wikipedia.org/wiki/Straw_man

H-connected, is continuous as Euler describes it, as Nunez and Lakoff
put it: "naturally continuous". As well I described various other
contexts where the function would be among continuous functions, in
analytical frameworks where said continuity is a useful property, for
example topologically with the novel contrapositive definition given
above.

This is mathematics, defend the argument, don't attack the man. And,
don't line up his effegies you made, beat them down, and claim
victory: that is fallacious argument.

Straw-men don't much resist.

You'll look above and find that I don't even contradict the definition
of continuity, instead that, where yes that function does meet the
definition of naturally continuous, and that's what I would call
continuous in natural language, that I'm happy to use that.

Euler's definition of "continuous", ancient as it is and now called
"naturally continuous", has this Heaviside's step with the riser:
being so.

Now, there are other definitions, of real analysis, even taught in
undergraduate real analysis today, of continuity.

Hardy: "It is natural to call a function continuous if its graph is a
continuous curve, .... Let us take this as a provisional definition
and try to distinguish more precisely some of the properties which are
involved in it."

(f(x) is defined, neighborhoods left and right defined, f(x-) = f(x)
= f(x+))

"The properties thus defined are far from exhausting those of a
picture of a curve by the eye of common sense, a picture which is a
generalization from particular curves such as straight lines and
circles. But they are the simplest and most fundamental properties:
and the graph of any function which has these properties would, so far
as drawing it is practically possible, satisfy our geometrical feeling
of what a continuous curve should be."

So, also Hardy agrees that there are naturally continuous functions,
beyond a regular working definition. The regular working definition,
so restricted, enables the general categorization of function into
that definition, then for the simple establishment of concomitant
analytical properties, but: it doesn't define all that "naturally
continuous" would be.

So, have at him: Hardy. He sees it: the provisional definition as
meets our intuition and the plain statement, and, one suitable for
wide application, with somewhat less generality.

Plainly, circles are continuous curves, and so is the step with riser.

http://en.wikipedia.org/w/index.php?title=Continuous_path&redirect=no

Regards,

Ross Finlayson



Date Subject Author
11/25/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
11/25/12
Read Re: Cantor's first proof in DETAILS
Virgil
11/25/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
11/25/12
Read Re: Cantor's first proof in DETAILS
Graham Cooper
11/25/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
11/25/12
Read Re: Cantor's first proof in DETAILS
Virgil
11/25/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
11/26/12
Read Re: Cantor's first proof in DETAILS
Virgil
11/26/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
11/26/12
Read Re: Cantor's first proof in DETAILS
Virgil
11/26/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
11/26/12
Read Re: Cantor's first proof in DETAILS
Virgil
11/27/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
11/27/12
Read Re: Cantor's first proof in DETAILS
Virgil
11/27/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
11/27/12
Read Re: Cantor's first proof in DETAILS
Virgil
11/27/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
11/28/12
Read Re: Cantor's first proof in DETAILS
Virgil
11/28/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
11/28/12
Read Re: Cantor's first proof in DETAILS
Virgil
11/28/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
11/28/12
Read Re: Cantor's first proof in DETAILS
Virgil
11/29/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
11/29/12
Read Re: Cantor's first proof in DETAILS
Virgil
11/29/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
11/30/12
Read Re: Cantor's first proof in DETAILS
Virgil
11/30/12
Read Re: Cantor's first proof in DETAILS
FredJeffries@gmail.com
11/30/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
11/30/12
Read Re: Cantor's first proof in DETAILS
FredJeffries@gmail.com
11/30/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
12/1/12
Read Re: Cantor's first proof in DETAILS
Virgil
11/30/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
12/1/12
Read Re: Cantor's first proof in DETAILS
Virgil
12/1/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
12/1/12
Read Re: Cantor's first proof in DETAILS
Virgil
12/1/12
Read Re: Cantor's first proof in DETAILS
FredJeffries@gmail.com
12/1/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
12/1/12
Read Re: Cantor's first proof in DETAILS
Virgil
12/2/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
12/2/12
Read Re: Cantor's first proof in DETAILS
Virgil
12/2/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
12/2/12
Read Re: Cantor's first proof in DETAILS
Virgil
12/3/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
12/3/12
Read Re: Cantor's first proof in DETAILS
Virgil
12/3/12
Read Re: Cantor's first proof in DETAILS
Virgil
12/3/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
12/3/12
Read Re: Cantor's first proof in DETAILS
Virgil
12/3/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
12/4/12
Read Re: Ross' Delusions re his EF.
Virgil
12/4/12
Read Re: Ross' Delusions re his EF.
ross.finlayson@gmail.com
12/4/12
Read Re: Ross' Delusions re his EF.
Virgil
12/4/12
Read Cantor's first proof in DETAILS
ross.finlayson@gmail.com
12/4/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
12/4/12
Read Re: Cantor's first proof in DETAILS
Virgil
12/4/12
Read Re: Cantor's first proof in DETAILS
Virgil
12/4/12
Read Re: Cantor's first proof in DETAILS
Virgil
12/5/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
12/6/12
Read Re: Cantor's first proof in DETAILS
Virgil
12/7/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
12/7/12
Read Re: Cantor's first proof in DETAILS
Virgil
12/8/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
12/9/12
Read Re: Cantor's first proof in DETAILS
Virgil
12/9/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
12/9/12
Read Re: Cantor's first proof in DETAILS
Virgil
12/9/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
12/9/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
12/9/12
Read Re: Cantor's first proof in DETAILS
Virgil
12/9/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
12/9/12
Read Re: Cantor's first proof in DETAILS
Virgil
12/9/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
12/9/12
Read Re: Cantor's first proof in DETAILS
Virgil
12/9/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
12/10/12
Read Re: Cantor's first proof in DETAILS
Virgil
12/10/12
Read Re: Cantor's first proof in DETAILS
12/4/12
Read Re: Cantor's first proof in DETAILS
Virgil
12/5/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
12/5/12
Read Re: Cantor's first proof in DETAILS
Virgil
11/30/12
Read Re: Cantor's first proof in DETAILS
Virgil
11/25/12
Read Re: Cantor's first proof in DETAILS
Graham Cooper
11/25/12
Read Re: Cantor's first proof in DETAILS
Virgil
11/25/12
Read Re: Cantor's first proof in DETAILS
ross.finlayson@gmail.com
11/25/12
Read Re: Cantor's first proof in DETAILS
Virgil

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