Re: Limits for functions defined on arbitrary sets (not intervals)

From: Keith A. Lewis (lewis_at_SPYDER.MITRE.ORG)
Date: 12/01/04 

Date: Wed, 1 Dec 2004 23:45:08 +0000 (UTC)

B Thomas <balbirthomas@hotmail.com> writes in article <Drrrd.25417$MG3.2226@fe2.columbus.rr.com> dated Wed, 01 Dec 2004 22:19:15 GMT:
> (a) For each posative number delta there is at least one point x in S
> with |x-a| < delta
> (b) For each posative number epsilon there is a posative number delta
> such that if |x-a|< delta and x in S, then |g(x)-a|< epsilon

>(2)
>On the other hand consider the function g(x)=x for all x in natural
>numbers except 1, and g(1)=2. We would like a definition that yeilds
>that the limit of g(x) as x tends to 1 is 1. However the above
>definition implies that this limit is 2.

That doesn't make sense to me. You can certainly interpolate (g(0)+g(2))/2
but that isn't the same as a limit.

The one case where a limit on a function which is in a discrete domain would
make sense is a=oo. Then you can test successively closer deltas for
validation. For example:

lim(x->oo) n!/(n-1)!/(n+1)

The epsilons are then infinite, but other than that it works.

--Keith Lewis klewis {at} mitre.org
The above may not (yet) represent the opinions of my employer.

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