I recall that, if $\psi:\Bbb{R}\longrightarrow \Bbb{R}$ is a function defined over $\Bbb R$ with euclidean topology, we have
$\liminf\limits_{y\to x} \psi(y) = \sup\limits_{U\in \mathscr{U}_x} \inf\limits_{y\in U\setminus \{x\}} g(y)$
where I called $\mathscr{U}_x$ the filter of neighborhoods at point $x$.
My goal is to prove that the function $f:\Bbb{R}\longrightarrow\Bbb{R}$ defined as
$f(x) = \liminf\limits_{y\to x} g(y)$
is lower semicontinuous, that is the set $L_f(t)=\{x\in \Bbb{R}\mid f(x)>t\}$ is open for every $t\in \Bbb{R}$. I noted that this is false if the topology over $\Bbb R$ is not Kolmogoroff or worse $T_1$, but that seems true for topologies with a decent grade of separation.