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Topic: The ambiguity of 0^0 on N
Replies: 106   Last Post: Sep 29, 2013 10:06 AM

 Messages: [ Previous | Next ]
 LudovicoVan Posts: 4,165 From: London Registered: 2/8/08
Re: The ambiguity of 0^0 on N
Posted: Sep 21, 2013 12:15 PM

"Michael F. Stemper" <michael.stemper@gmail.com> wrote in message
news:l1kfqh\$gde\$1@dont-email.me...
> On 09/20/2013 04:48 PM, Julio Di Egidio wrote:
>> "Michael F. Stemper" <michael.stemper@gmail.com> wrote in message
>> news:l1ie0g\$rpg\$1@dont-email.me...

>>> On 09/20/2013 02:42 PM, Virgil wrote:
>
>>>> I can see that
>>>> f(x,y) = x^y is continuous on domain {(x,y): x > 0 and y > 0}
>>>> and can even be extended continuously to both
>>>> {(0,y): y > 0} and
>>>> {(x,0): x > 0}
>>>> but there is no way to extend that function CONTINUOUSLY to
>>>> {(x,y)| x > 0 and y > 0} to {(x,y)| x >= 0 and y >= 0}.
>>>>
>>>> This suggest that for real functions like
>>>> f(x,y) = x^y on {(x,y): x > 0 and y > 0}
>>>> there is no universally acceptable limit value at f(0,0).

>>>
>>> That's true for the reals, yes. I believe that the topic under
>>> discussion is "N", the naturals.

>>
>> Aren't the naturals a subset of the reals?

>
> Well, they're isomorphic to a subset of the reals, anyway.

Right.

> However, what metric on N would you use that would give a
> singular, well-defined limit of any function at any point?

I am on the side that thinks 0^0 must be undefined in (a point-like number)
arithmetic.

Here is my *proof*, using the usual rules:

0^0
= 0^(1-1)
= (0^1)/(0^1)
= 0/0
= undefined

Julio

Date Subject Author
9/18/13 Dan Christensen
9/18/13 Peter Percival
9/18/13 Dan Christensen
9/18/13 Peter Percival
9/18/13 Virgil
9/18/13 Dan Christensen
9/18/13 Rotwang
9/18/13 Rock Brentwood
9/18/13 Rotwang
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Dan Christensen
9/20/13 fom
9/19/13 Virgil
9/19/13 Virgil
9/19/13 Rotwang
9/18/13 Virgil
9/18/13 fom
9/18/13 Rotwang
9/28/13 Shmuel (Seymour J.) Metz
9/29/13 Marshall
9/19/13 Dan Christensen
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 Michael F. Stemper
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Dan Christensen
9/20/13 fom
9/20/13 Dan Christensen
9/20/13 fom
9/19/13 fom
9/19/13 fom
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Rotwang
9/19/13 Dan Christensen
9/19/13 Helmut Richter
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 fom
9/19/13 JT
9/19/13 JT
9/19/13 Michael F. Stemper
9/19/13 JT
9/19/13 JT
9/19/13 JT
9/19/13 Helmut Richter
9/28/13 Shmuel (Seymour J.) Metz
9/19/13 fom
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Karl-Olav Nyberg
9/19/13 fom
9/19/13 fom
9/19/13 Rotwang
9/19/13 Dan Christensen
9/19/13 fom
9/25/13 Rotwang
9/26/13 Dan Christensen
9/27/13 Brian Q. Hutchings
9/19/13 fom
9/18/13 Rock Brentwood
9/19/13 Dan Christensen
9/19/13 Dan Christensen
9/19/13 Rotwang
9/19/13 Dan Christensen
9/19/13 fom
9/20/13 Dan Christensen
9/20/13 fom
9/20/13 Dan Christensen
9/20/13 Peter Percival
9/20/13 Peter Percival
9/20/13 Dan Christensen
9/20/13 Virgil
9/20/13 Peter Percival
9/20/13 fom
9/20/13 Michael F. Stemper
9/20/13 LudovicoVan
9/21/13 Michael F. Stemper
9/21/13 LudovicoVan
9/21/13 Richard Tobin
9/20/13 Peter Percival
9/20/13 Peter Percival
9/21/13 Dan Christensen
9/19/13 Karl-Olav Nyberg