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When I read Serge Lang's Undergraduate Analysis(second edition) page 41, I found the definition of limit he define is so called non-Deleted limits, this results in some conclusion is different from the usual limit( deleted limit), for example, every isolated point $x_0$ has limit, its value is $f(x_0)$. By topology knowledge, we know every isolated point is continuous, we cannot get this result by $\lim_{x\to x_0}f(x)=f(x_0)$, if we define limit as deleted limit. However, if we define limit as non-deleted limit, we can get it.

My question: is non-deleted limit better than deleted limit, if yes, why does most textbook do not use it?

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  • $\begingroup$ I think the existence of non-deleted limits is more or less equivalent to continuity. $\endgroup$ – Vim May 22 '16 at 3:47
  • $\begingroup$ The two concepts agree for $x \to x_0$ provided $x_0$ happens not to be in the domain of $f.$ But if $x_0$ is in fact in the domain of $f$ the nondeleted limit existence implies continuity of $f$ at $x_0.$ $\endgroup$ – coffeemath May 22 '16 at 6:54
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A good place to get information on deleted and non-deleted limits is Robert G. Bartle's Elements of Real Analysis, J. Wiley & Sons 1964, p. 195

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