Date: Dec 9, 2012 6:23 PM Author: ross.finlayson@gmail.com Subject: Re: Cantor's first proof in DETAILS On Dec 9, 2:47 pm, Virgil <vir...@ligriv.com> wrote:

> In article

> <ec1e904c-767e-44fa-a13e-21e38f605...@jj5g2000pbc.googlegroups.com>,

> "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote:

>

> > 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.

>

> Then it is your straw man.

>

> Since you failed to demonstrate that being "naturally continuous"

> differs in your mind, or anyone else's, from naturally being

> "continuous", it is you who are in the wrong here, intending to deceive.

>

> And you also failed to give any definition of being "naturally

> continuous", or any reason to suspect that it had its own meaning.

>

> So I repeat, your original claim of continuity was wrong, and until yo

> provide your definition of "naturally continuous", you claim of a

> discontinuous function being "naturally continuous" is still not

> exanblished.

> --

Shoo, fly.

"There's no gap in this Heaviside step with connecting H(0+) and

H(0-)

with a simple line segment. There is no point in it such that, not

in

the function, it is the only point in all neighborhoods of any [1]

two

points in the function, not in the function (not even a point

discontinuity). Here "in the function" is each (x,y) in the combined

coordinate image or co-range, with the function defined by the points

in it. The two points are from: the left and right limit

sequences,and

the points on the asymptote. (The contrapositive is a strong

rationale not all find.) " ([1] of every)

That one's mine: a definition of continuity. I'd well surmise it's

already found, but, I discovered it.

However again as noted, though, you're barking up Euler's and Hardy's

trees, who we hold in high esteem and of authority. Hardy's

apologetics as noted above may help, as his treatise was the text.

Euler and Hardy give definitions of "naturally continuous", for what

we know that is yet, somewhat, under-defined, reflected in the

"natural" as being "fundamental", "defining", or "primary".

And then, no, I'm quite against any "attempt to deceive": quite.

Hancher, I think you should give our readers somewhat more credit in

terms of their rational ability, else, they give you less. Also I'd

be glad to see that in this thread already you've corrected your

statements that Aleph_0 is in the reals and H isn't a real function.

Dear readers, I think very highly of you. Then warm regards in the

spirit of the season and good luck with your reflections on

mathematical: truth.

And warm regards,

Ross Finlayson