Find a sequence of continuous functions on $[0,1]$, $\{f_n\}$, such that $f(x):=\sup\{f_n(x):n\geq 1\}$ and $g(x):=\inf\{f_n(x):n\geq 1\}$ are both not continuous.
I kept finding examples where one was discontinuous but the other was continuous like $f_n(x)=|x|^\frac{1}{n}$ and $f_n(x)=x^n$