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.

Solution 1:

It is defined as

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