Semicontinuous Functions

Let the function $f$ be defined on $\mathbb {R}$. Prove that the following properties of the function $f$ are equivalent (compare with Problem 1.2):

1. the function $f$ is lower semicontinuous;

2. the set $f^{-1}((c,+\infty ))$ is open for every $c \in \mathbb {R}$;

3. the set $f^{-1}((-\infty , c])$ is closed for every $c \in \mathbb {R}$;

4. if $x_{n} \rightarrow a$, then $f(a) \leqslant \sup f\left(x_{n}\right)$;

5. for every point $a \in \mathbb {R}$ and every number $\varepsilon >0$ there exists a neighborhood $V$ of the point $a$ such that $f(x) \geqslant f(a)-\varepsilon$ for $x \in V$.

Prove that

Prove that if a function is lower semicontinuous on the segment $\left[a, b\right]$ and takes only finite values, then it attains its least value on $\left[a, b\right]$ (and hence is bounded below). Is it true that it is bounded above?

Let $g$ be an arbitrary function defined on an interval $\Delta \subset \mathbb {R}, f(x)=\varliminf_{y \rightarrow x} g(y)(x \in \Delta )$. Prove that the function $f$ is lower semicontinuous.

Let $\left\{ f_{\alpha }\right\}_{\alpha \in A}$ be a family of lower semicontinuous functions defined on an interval $\Delta \subset \mathbb {R}, f(x)=\sup_{\alpha \in A} f_{\alpha }(x)$ and $f(x)<+\infty$ for every $x \in \Delta$. Prove that the function $f$ is lower semicontinuous. Is this true for the function $h=\inf_{\alpha \in A} f_{\alpha }$? Will the function $h$ be lower semicontinuous if the family $\left\{ f_{\alpha }\right\}_{\alpha \in A}$ is finite? countable?

Prove that the function $f:\left[a, b\right] \rightarrow \mathbb {R}$ is lower semicontinuous if and only if