On Mar 5, 7:40 am, "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote: > On Mar 4, 8:08 pm, William Elliot <ma...@panix.com> wrote: > > > On Mon, 4 Mar 2013, Ross A. Finlayson wrote: > > > Does n/d have these simple analytical properties? > > > Yes, that's a verbose characteristic of f_d which I [think] > > you're talking about and not some arbitrary function f. > > Thank you, yes, that's so. >
Rather leading questions, obviously enough anyone familiar with my extended writings knows this is Socratic about the "natural/unit equivalency function."
Is there any neighborhood in [0,1] not containing infinitely many points in the range of f? No, the points in the range of f are dense in [0,1].
Then this simple construct of a function modeled by a family of functions, "ranges" from zero to one, with that over the domain the "linear" function sees values in the range between and including zero and one and corresponding to the natural order of the domain, that "ranges" doesn't need quotes, f ranges from zero to one.
Then, those are some simple properties of this construct. Then, the free mathematical mind simply enough considers conditions as of other analytical properties of this function. With n e R, or x e R+ for f_d(x), Int_0->d f_d(x) dx = d/2. This is simply enough the area under the line connecting the origin and (d,1). How about for n e R? It, for r e ran(f_d(n)), f_d(rd), looks like f(x) = x, from zero to one, the simple ray of a line with unit slope, from zero to one. The area under that is 1/2. But, the "area" under f_oo(n) = 1. Consider lim_d->oo Int_0->d 1/d dx, that equals one. This looks like a flat line infinitesimally greater than zero, the area under which sums to one. So, there are some _interesting_ properties, of f_d(n) = n/d.