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To understand the notion of, say, limit superior for a sequence, is not difficult. Simply consider the set of all upper buonds for the set of all limit points of the sequence, and then simply pick the inf of this set. I said simply, because for a sequence, we only take limits to infinity.

Now, for a function we can calculate limits to any point.

So, what exactly means $$\limsup_{x\to c} f(x) $$

Wikipedia is unclear about this, and I found nothing in my (tiny) literature.

Thanks in advance.

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The supremum is the least upper bound on $f(x)$. Therefore by definition of limit we have that it is least upper bound on a point. I think that it only exists if $\textup{lim} \ \ textup{sup} f(x)= \textup{lim} \ \textup{sup} f(c)$. –  Jaivir Baweja Jan 25 '13 at 0:02

1 Answer 1

up vote 2 down vote accepted

It is defined as

$$ \underset{x \to a}{\mbox{lim sup}} f(x) = \lim_{\epsilon \to 0^{+}} (\sup \{f(x) : x \in B(a,\epsilon) \setminus \{a\}\})$$

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But if epsilon tends to zero, this appears to be precisely the regular limit... –  MadHatter Jan 25 '13 at 0:02
    
If the limit exists, this will give the same value. But the lim sup exists even if the limit does not. For example, consider the function that is $1$ on the rationals and $0$ on the irrationals. The lim sup of this function as you approach $0$ is $1$, and the lim inf is $0$, but no limit exists. –  Isaac Solomon Jan 25 '13 at 0:04
    
I'm starting to understand. Can you give me some example involving semicontinuity? –  MadHatter Jan 25 '13 at 0:05
    
Can you clarify that? What do you mean by semicontinuity, I don't see how general semicontinuity relates to this. –  Isaac Solomon Jan 25 '13 at 0:07
    
See for example en.wikipedia.org/wiki/Semi-continuity#Formal_definition. Thanks again –  MadHatter Jan 25 '13 at 1:37

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