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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?

noname1014
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  • I think the existence of non-deleted limits is more or less equivalent to continuity. – Vim May 22 '16 at 03:47
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    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 $x_0.$ – coffeemath May 22 '16 at 06:54

2 Answers2

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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

Augusto Oliveira
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Hubbard and Hubbard's Vector Calculus, Linear Algebra, and Differential Forms claims that deleted limits are standard in the U.S. but non-deleted limits are common in France. One thing that is nice about deleted limits is that they are sometimes defined when non-deleted limits aren't. In the book, they use non-deleted limits because it makes limits much more well-behaved under compositions. Specifically, if $$y_0 = \lim_{x \to x_0} f(x) \quad \textrm{and} \quad z_0 = \lim_{y \to y_0} g(y)$$ both exist, then $\lim_{x \to x_0} g(f(x))$ exists and $$\lim_{x \to x_0} g(f(x)) = z_0.$$ This is not true for deleted limits.

Math2718
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