Re: There is a first/smallest integer (in Mückenland)

On 7/18/2024 1:52 AM, Moebius wrote:

Am 18.07.2024 um 06:30 schrieb Jim Burns:

On 7/17/2024 1:49 PM, WM wrote:

Le 17/07/2024 à 19:13, FromTheRafters a écrit :

Sure,
it jumps because of your stepwise function.

>
Of course it jumps,
but what is the maximum size of a jump?

>
|ℝ| is the maximum size of a jump.

>
Nope.

You are answering a different question,
| What is the size of the jump of NUF(x) at 0?
I had read "a jump" as a reference more generic than that.
-- However, in WM's most recent post, it seems that
your reading is more correct than mine.
Paraphrasing, WM seemed to ask
| How can a jump at one point be by
| more than one point? Anywhere. Any jump.
Paraphrasing you
| Well, it _is_ more.
| <same proof again>
Paraphrasing me
| Here is how.
| A jump _can_ be by
| much more than it _is_ in NUF(x) at 0
| <different conceivably.understood proof>
|ℝ| is the maximum size of a jump.
∀ᴿx≤0: NPR(x) = |(0,x)| = |{}| = 0
∀ᴿx>0: NPR(x) = |(0,x)| = |ℝ|
f(y) = y/√​̅y​̅²​̅+​̅1
f: ℝ → (-1,1): 1.to.1
g(y) = (y+1)/2
g: (-1,1) → (0,1): 1.to.1
g∘f(y) = (y/√​̅y​̅²​̅+​̅1+1)/2
g∘f: ℝ → (0,1): 1.to.1
x⋅g∘f(y) = x⋅(y/√​̅y​̅²​̅+​̅1+1)/2
x⋅g∘f: ℝ → (0,x): 1.to.1
|ℝ| ≤ |(0,x)|
Also
ℝ ⊇ |(0,x)|
|ℝ| ≥ |(0,x)|
|ℝ| = |(0,x)|
∀ᴿx>0: NPR(x) = |(0,x)| = |ℝ|
Jumps "at" a point are between
  nearby points.
WM admitted that much in a recent post,
but changed what "change" means to him.

|IN| is "the maximum size of a jump",
namely "the size" of the jump "at" 0.
(Hint: NUF has only one jump, namely "at" 0.)

I continue to accept the proofs that
∀ᴿx>0: NUF(x) = |⅟ℕ∩(0,x)| = |⅟ℕ|