# Lim Sup/Inf for real valued functions

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

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