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

From: B.Thomas (balbirthomas_at_hotmail.com)
Date: 12/02/04 

Date: Thu, 02 Dec 2004 04:27:32 GMT

Keith A. Lewis <lewis@SPYDER.MITRE.ORG> wrote:
> B Thomas <balbirthomas@hotmail.com> writes in article :
>> (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.
>
No I am not interpolating. Just using the definition as in a
epsilon-delta proof of a limit to show the limit is indeed 2. However if
the domain was real numbers then the limit would be 1, for the same
function. Which is why I said "we would like a definition that yeilds
" 1 as the 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.

One can talk of continuity in arbitrary topological spaces. And if the
topology is induced by a metric then we can certainly talk about limits
even if we are dealing with point sets and not intervals.

sincerely
b thomas

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