What can one say about the set of points where a function is lower semicontinuous?

Question

Given a function $f:X\to\overline{\mathbb{R}}$ what can one say about the points where $f$ is lower semicontinuous, that is, about $$ M:=\{x\in X\mid f{\rm\ is\ lower\ semicontinuous\ at\ }x\}? $$

Is it open, closed, $G_\delta$, $F_\sigma$, etc.?

Here $(X,\|\cdot\|)$ is a normed space and $f$ is lower semicontinuous at $x$ if $\forall\epsilon>0$, $\exists\delta_\epsilon>0$, $\forall y\in X$ with $\|y-x\|\le\delta_\epsilon$ one has $f(x)\le f(y)+\epsilon$.

Answer 1

The characteristic function $\chi_{\mathbb{Q}}(x):\mathbb{R}\to \overline{\mathbb{R}}$ will show you that in general $M$ is neither open nor closed. In this case $M= \mathbb{Q}$. This also says that $M$ is not in general $G_\delta$.

On the other hand, the characteristic function $\chi_{\mathbb{R}\setminus \mathbb{Q}}(x):\mathbb{R} \to \overline{\mathbb{R}}$ has $M=\mathbb{R}\setminus\mathbb{Q}$, which is not $F_\sigma$.