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From: Moebius <invalid@example.invalid>
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Subject: =?UTF-8?Q?Re=3A_There_is_a_first/smallest_integer_=28in_M=C3=BCcken?=
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Date: Thu, 18 Jul 2024 08:00:07 +0200
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Am 18.07.2024 um 07:52 schrieb Moebius:
> 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. |IN| (aleph_0) is "the maximum size of a jump", namely "the size" of the 
> jump "at" 0. (Hint: NUF has only one jump, namely "at" 0.)

Hint:   NUF(x) := |{u e {1/n : n e IN} : u <= x}| (x e IR) .

Hence

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

as well as

	img(NUF) = {0, aleph_0} .