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From: Moebius <invalid@example.invalid>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions? (infinitary)
Date: Sat, 2 Nov 2024 14:03:16 +0100
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Am 02.11.2024 um 13:30 schrieb joes:
> Am Fri, 01 Nov 2024 11:07:27 +0100 schrieb WM:
>> On 31.10.2024 21:46, joes wrote:
>>
>>> At single points a function has a single value, not a jump.
>> It jumps in case of NUF by 1 at a unit fraction with respect to the
>> foregoing unit fraction and the many points between both.
> Whether you define your pathological function with < or <=, at every
> point it has one value (note it doesn't have the SAME value at every
> point [...]).
> 
>>> In the case of NUF, [we have:]

NUF(x) = 0 for x = 0 and
NUF(x) = aleph_0 for all x e IR, x > 0.

Instead of NUF we might consider the function

      NNN(x) = card {n e IN: n > x}    (x e IR*)

      "number of natural numbers larger than x"

defined on the extended reals (R* = IR u {-oo, oo}).

THERE we would have:

NNN(x) = 0 for x = oo.
NNN(x) = aleph_0 for all x e IR*, x < oo.

Math is too hard for Mückenheim.