Upper semi continuous, lower semi continuous which of the followings are true?


*

*$X$ be a topological space,  $f_n:X\rightarrow \mathbb{R}$ is sequence of lower semi continuous functions then the $\sup\{f_n\}=f$ is also lower semi continuous.

*every continuous real valued function on $X$ is lower semi continuous.

*A real valued function on $X$ is continuous iff it is both USC and LSC.


I read in my measure theory course and recall that $3$ and $1$ is true though I can not remember the proofs now, but could any one just give me hint how to handle $2$? Thank you.
 A: One definition that can be used for $\small\begin{array}{c}\text{upper}\\\text{lower}\end{array}$-semicontinuity is that $f$ is $\small\begin{array}{c}\text{upper}\\\text{lower}\end{array}$-semicontinuous if and only if
$$
\{x:f(x)\lessgtr\alpha\}
$$
is open for all $\alpha$.
Hints


*

*Note that
$$
\{x:\sup_{n\ge1}f_n(x)\gt\alpha\}=\bigcup_{n=1}^\infty \{x:f_n(x)\gt\alpha\}
$$

*One definition that can be used for continuity is that $f$ is continuous if and only if $f^{-1}(U)$ is open for all open $U$. Then note that $\{x:f(x)\gt\alpha\}=f^{-1}\left(\{x:x\gt\alpha\}\right)$.

*In a fashion similar to 2. we can show that every continuous function is upper-semicontinuous. Thus, we just need to show that each function that is both upper and lower semicontinuous is continuous. Suppose that $f$ is both upper and lower semicontinuous. Then
$$
f^{-1}(\alpha,\beta)=\{x:f(x)\gt\alpha\}\cap\{x:f(x)\lt\beta\}
$$
is open for all $(\alpha,\beta)$. Furthermore, for every open set, $U$,
$$
U=\bigcup_{u\in U}(u-\epsilon_u,u+\epsilon_u)
$$
where $\epsilon_u\gt0$ is chosen so that $(u-\epsilon_u,u+\epsilon_u)\subset U$.
