Is There a New Theorem in Limits Without Accumulation Points?

[poutsos.A]

possible theorem in limits??

Is the following a possible theorem in limits??


If a is not an accumulation of the domain of...$$f:A\subseteq R\rightarrow R$$ then f has a limit over a

 

[HallsofIvy]

On the contrary, in order for a function to have a limit at a, a MUST be an accumulation point of the domain!

Did you mean to write:
"If a is not an accumulation of the domain of...$$f:A\subseteq R\rightarrow R$$ then f does not have a limit at a"?

 

[poutsos.A]

On the contrary, in order for a function to have a limit at a, a MUST be an accumulation point of the domain!

Did you mean to write:
"If a is not an accumulation of the domain of...$$f:A\subseteq R\rightarrow R$$ then f does not have a limit at a"?

Why can you not have a limit provided the definition of a limit of a function at a point is the following:


Let:

1) $$f:A\subseteq R\rightarrow R$$

2) a is adherent to A ......

..........THEN we define.........

......limf(x)= m ,x----->a iff $$\forall \epsilon[\epsilon>0\rightarrow\exists r(r>0 ,\forall x( x\epsilon D(f)\wedge |x-a|<r\rightarrow |f(x)-m|<\epsilon))]$$

And in words :

......limf(x) = m ,x------->a iff given ε>0 there exists r>o such that if


...... xεD(f) & |x-a|<r then |f(x)-m|<ε..........

D(f)= domain of f

 

[HallsofIvy]

I don't see how that would be a very useful definition. Using that definition, if a is not an accumulation point of the domain of f, then "$\lim_{x\rightarrow a} f(x)= L" is true for EVERY L. It would be incorrect to talk about "<b>the</b> limit at a" when every number is a limit at a.$

 

[poutsos.A]

If a is not accumulation point of the domain of f then a is an isolation point and we have to consider two cases:

case 1: a belongs to the domain of f: in this case the said definition gives a limit for every function which is f(a)

case 2 a does not belong to the domain of f then a is not an inherent point of THE domain and the said definition is not applicable:


But that is not what i Had in mind. The theorem i had in mind to be proved is:

If a is not an accumulation point of the domain of the function:$$f:A\subseteq R\rightarrow R$$ then there exists m such that:


........given ε>0 there exists r>0 such that.......


......if xεA and 0<|x-a|<r then |f(x)-m|<ε.......

 

[poutsos.A]

The above definition was given just to show that there can be definitions other than the usual one which uses the accumulation point .THE above definition covers awide range of functions