Definition
2
[2GW] Let \(E\subseteq X\), \(x_ 0∈ X\) accumulation point of \(E\), \(f:E→ ℝ\) function. We define
\begin{align} \limsup _{x→ x_ 0} f(x) = \inf _{U \text{~ neighbourhood of~ } x_ 0}~ \sup _{x∈ U∩ E} f(x) \label{eq:limsup_ X}\\ \liminf _{x→ x_ 0} f(x) = \sup _{U \text{~ neighbourhood of~ } x_ 0}~ \inf _{x∈ U∩ E} f(x) \label{eq:liminf_ X} \end{align}
where the first ”inf” (resp. the ”sup”) is performed with respect to the family of all the deleted neighbourhoods \(U\) of \(x_ 0\). 1