```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+) andH(0-)with a simple line segment.  There is no point in it such that, notinthe function, it is the only point in all neighborhoods of any [1]twopoints in the function, not in the function (not even a pointdiscontinuity).  Here "in the function" is each (x,y) in the combinedcoordinate image or co-range, with the function defined by the pointsin it.  The two points are from: the left and right limitsequences,andthe points on the asymptote.  (The contrapositive is a strongrationale not all find.) "  ([1] of every)That one's mine: a definition of continuity.  I'd well surmise it'salready found, but, I discovered it.However again as noted, though, you're barking up Euler's and Hardy'strees, who we hold in high esteem and of authority.  Hardy'sapologetics as noted above may help, as his treatise was the text.Euler and Hardy give definitions of "naturally continuous", for whatwe 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 interms of their rational ability, else, they give you less.  Also I'dbe glad to see that in this thread already you've corrected yourstatements 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 thespirit of the season and good luck with your reflections onmathematical:  truth.And warm regards,Ross Finlayson
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