Re: Does the number of nines increase?

On 7/8/2024 9:49 AM, WM wrote:

Le 08/07/2024 à 01:57, Jim Burns a écrit :

On 7/7/2024 4:04 PM, WM wrote:

The question is only this:
Are there more than one unit fractions in
the point where NUF(x) changes from 0 to more,
or is there only one unit fraction?

>
The complete distribution of unit fractions,

>
does not answer this question.

...determines which things you are asking about.
Are you asking about things for which
each nonempty set holds a rightmost element (nearest 1)?
Are you asking about things for which
each u (including 1) has ⅟(1+⅟u) next.thing.to.the.left?
Are you asking about things for which
each v (excluding 1) has ⅟(-1+⅟v) next.thing.to.the.right?
If your answers are yes, yes, and yes,
then, whatever the things are called,
the answer to your question is the third option: none.
/ If there is a point x > 0: NUF(x) < ℵ₀
| then there is a leftmost unit.fraction β in (0,x]
| and unit fraction u₂ᵦ < 2⋅β not.leftmost
| and unit.fraction ¼⋅u₂ᵦ < ½⋅β < β
| to the left of leftmost unit.fraction β
\ Contradiction.
There is no point x > 0: NUF(x) < ℵ₀
There is no point x < 0: NUF(x) > 0
NUF(x) changes "at" 0.
And,
if any of the three answers is no,
then you're asking about something else.
Maybe some day you'll say what you're asking about.

NUF(0) = 0
NUF(x) changes at 0
There is no unit fraction at 0

>
Therefore it does not change at 0.

Functions do not change at single points.
A change needs be _with respect to_ something,  and
the value at a point with respect to itself
is unchanging.
Consider the function
NP(x) = |{.01,.02,.03,…,1.00}∩(0,x]|
NP(.005) = 0
NP(.015) = 1
NP(x) changes between .005 and .015
for 0<β<.01
NP(.01-β) = 0
NP(.01+β) = 1
NP(x) changes between .01-β and .01+β
NP(.01) = NP(.01) is unchanging.
Consider once more the function
NUF(x) = |⅟ℕ∩(0,x]|
for each x
NUF(x) = NUF(x) is unchanging.
for 0<β
NUF(0-β) = 0
NUF(0+β) = ℵ₀
NUF(x) changes between 0-β and 0+β
for 0<β
NUF(0+β) = ℵ₀
because, otherwise,
a unit fraction u₂ᵦ exists such that
  ¼⋅u₂ᵦ is NOT a unit fraction.