Upper semi continuous, lower semi continuous

Solution 1:

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

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

2. 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 $\{x:f(x)\gt\alpha\}=f^{-1}\left(\{x:x\gt\alpha\}\right)$.

3. 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 $(u-\epsilon_u,u+\epsilon_u)\subset U$.