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. > --
"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  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.) " ( 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.