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In the textbook I'm studying (Convex Analysis ... by Bauschke and Combettes), liminf is defined below:

for $f:\mathcal{X} \rightarrow [-\infty, \infty], x\in \mathcal{X}$, define:

$\underset{y\rightarrow x}{\underline{\text{lim}}}f(y)=\underset{V\in\mathcal{V}(x)}{\text{sup}}\text{inf}\text{ }f(V)$, where:

$\mathcal{V}(x)$ is the set of all neighborhoods around $x$.

In wiki and most other sources I checked, the definition uses "deleted neighborhoods" $\mathcal{V}(x)\backslash\{x\}$ instead. And this would make a difference if we consider the function $f$ defined by: $f(x) = 2, \text{ }\forall x\ne 0$, and $f(0) = 1$.

According to the answer in this post, there are two definitions of limsup/liminf, one using deleted neighborhoods, the other using undeleted ones. But I wonder if there is any benefit of using the latter version?

TomG
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    According to the answer in this post --- More details about the two definitions are given in my comments to this MSE answer. – Dave L. Renfro Jan 13 '25 at 08:15
  • @DaveL.Renfro Thank you very much! Your comments are exactly the answer I'm looking for. – TomG Jan 13 '25 at 11:00
  • Actually, I thought you wanted to know whether there was any benefit to using undeleted neighborhoods, and I wasn't sure. Using undeleted neighborhoods allows for what could be seen as simpler versions of upper/lower semicontinuity definitions -- use equalities rather than inequalities, which makes the definitions better resemble continuity (have "limit = value", except use an extreme limit in place of "limit"). But doing this then makes the extreme limits less useful (or more awkward to use) in those situations where one is only concerned with behavior near (but not at) a boundary point. – Dave L. Renfro Jan 13 '25 at 14:08
  • @DaveL.Renfro I was too happy to confirm there are two versions of definitions I forgot my follow-up question. I believe you are right about that too -- it has something to do with usc/lsc statements. As another example, in my textbook, a follow-up lemma about lsc used liminf(f) <= f, which is only true using undeleted neighborhoods. – TomG Jan 14 '25 at 03:18

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