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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions? (infinitary)
Date: Wed, 30 Oct 2024 12:53:09 +0100
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On 29.10.2024 11:02, FromTheRafters wrote:
> WM formulated the question :
>> On 28.10.2024 21:51, FromTheRafters wrote:
>>> WM explained :
>>>> On 28.10.2024 15:07, FromTheRafters wrote:
>>>>> WM was thinking very hard :
>>>>
>>>>>> The function NUF(x) = Number of Unit Fractions between 0 and x 
>>>>>> starts with 0 at 0.
>>>>>
>>>>> Followed by a discontinuity.
>>>>
>>>> No. ∀n ∈ ℕ: 1/n - 1/(n+1) > 0 holds everywhere.
>>>
>>> Not for n equal to zero.
>>
>> I said ∀n ∈ ℕ. 0 is no natural number. NUF(0) = 0.
> 
> That is why your function is a step function with a discontinuity.

NUF(0) = 0, NUF(1) = ℵo. Therefore NUF must grow but cannot grow by more 
than 1 at any point x of the real axis.

Regards, WM