Just some quick notes (more for me to keep track of things to think about, rather than me trying to prove anything here). I think that given my previous two posts, -*! corresponds to the positive number line while +*! corresponds to the negative number line. That is, since zero is a reflecting point, and *! Is functionally equivalent to 0, then *! is also a reflecting point. However, if you try to picture this mentally, in order to keep positive integers from being cancelled out by the graph created at *!, you’d have to invert the signs in one of them.
Secondly, where functions are undefined (such as the tangent function wherever cos x = 0), I think that’s where the sign switches at the infinity mark. In other words, just as hitting zero on a number line switches our sign, so hitting *! switches the sign. This is sort of like saying the function actually is continuous, but in a different dimension that we cannot draw on the 2D grid. In other words, if we flip it and instead graph it from the x-axis crossing y at *!, then we’d see the line go through the point at *! and we’d be referring to the positive and negative zero, rather than the positive and negative infinity, and we’d be saying that the function is undefined at 1/n as n-> infinity (instead of how it is now, when it’s as n -> 0).
In other words, I think that the only reason these functions are considered undefined there is that we cannot visualize the two infinities meeting. Except that we can once we realize that it would look exactly like the zero line. We still couldn’t draw it, but we could at least visualize what it ought to look like, at least as far as we can mentally think of any infinite.
