Exercise
48
[29R]Prerequisites:[20F],[29J],[06M],[0FR],[0FT].Difficulty:*.(Proposed on 2022-11-24)
Let \(I⊂ ℝ\), \(x_ 0∈ \overline{ℝ}\) accumulation point of \(I\), \(f:I→ ℝ\) function. As in [29J] \({\mathcal F}\) all the neighbourhoods of \(x_ 0\) with associated the filtering ordering
\[ U,V∈ {{\mathcal F}}~ ~ , U≤ V \iff U⊇ V\quad . \]
Let
\[ s,i : {\mathcal F}→ℝ~ ~ ,~ ~ s(U) = \sup _{x∈ U∩ I} f(x)~ ~ ,~ ~ i(U) = \inf _{x∈ U∩ I} f(x) \]
note that these are monotonic functions, and show that 1
\begin{align} \limsup _{x→ x_ 0} f(x) {\stackrel{.}{=}}\inf _{U∈ {\mathcal F}} s(U) = \lim _{U∈ {\mathcal F}} s(U) \label{eq:limsup_ R_ F} \\ \liminf _{x→ x_ 0} f(x) {\stackrel{.}{=}}\sup _{U ∈ {\mathcal F}} i(U) = \lim _{U ∈ {\mathcal F}} i(U) \label{eq:liminf_ R_ F} \end{align}
where the limits are defined in [0FR].