Re: How many different unit fractions are lessorequal than all unit fractions? (infinitary)

On 02.11.2024 01:55, Richard Damon wrote:

On 11/1/24 6:07 AM, WM wrote:

On 31.10.2024 21:46, joes wrote:
>

At single points a function has a single value, not a jump.

>
It jumps in case of NUF by 1 at a unit fraction with respect to the foregoing unit fraction and the many points between both.

 If it jumps a *A* point, it has two values at that point,

No it has a larger value at that point than at the interval before that point.

In the
case of NUF, that value is infinite everywhere except at 0.

>
That is impossible because there are not infinitely many unit fractions between 0 and everywhere.
>

Why not?

If NUF(x) has grown to ℵ₀ at x₀, then ℵ₀ unit fractions must be between 0 and x₀. Hence at least ℵ₀ points with ℵ₀ intervals of uncountably many points must be between 0 and x₀. That cannot happen at x₀ = 0.
Regards, WM