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

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.

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

|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|