In article <email@example.com>, "Ross A. Finlayson" <firstname.lastname@example.org> wrote:
> 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  > 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.