On Dec 9, 7:41 pm, Virgil <vir...@ligriv.com> wrote: > In article > <72919023-5457-4eca-8d7c-31e93d640...@r10g2000pbd.googlegroups.com>, > "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote: > > > "There's no gap in this Heaviside step with connecting H(0+) and H(0-) > > with a simple line segment. > > Since the H jumps from 0 to 1 at x = 0, tha segeemnt must have endpoints > (0,0) and (0,1), > The fiction being claimed to be continuous, for what x does the function > take the value 1/2? >
> > 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. > > I do not find enough sense in that to be worth its refutation. > --
For the vertical line, for which x does the function take any value? Zero. Here now I'll call your reply: H-connected isn't a function, now, either, and neither are vertical lines, and then I'll refer you back to Euclid and Hardy so you can castigate them for conciliation, and in apologetics.
No, I well imagine you don't find enough sense that its refutation would be worth it (to you). Basically it says that, for any two points in the set of points defining the function, and every, if all their neighborhoods contain only one point, then the function contains that point if it is continuous (here, say, reductio continuous). Here it is that with a piecewise continuous function, that only what would be the point discontinuities need be analyzed that way, though it would seem to apply generally. As the defined points in the piecewise continuous elements go to their endpoint, more and more of their neighborhoods contain the points on the other side of the point gap. When all those neighborhoods contain a point, it is defined in the function (defining the function) for the function to be continuous. This is simply not far removed from our topological definitions of continuity, after density: gaplessness.
We've drifted somewhat afield from Cantor's first proof in details and here with regards to "A function surjects the rationals onto the irrationals" and "EF as a function has different results than any other in Cantor's first (and the antidiagonal argument)".