the sum of a sequence of lower semicontinuous functions may not be lower semicontinuous?

Question

Let $f_n$ be lower semicontinuous, and $f_n\geq 0$, then $\sum f_n$ is lower semicontinuous. Indeed, for any $c$, $\{x; \sum f_n(x)>c\}=\bigcup_{n=1}^\infty \{x; \sum_{k=1}^n f_k(x)>c\}$.

However, if $f_n\geq 0$ is deleted, I suspect it is wrong. But I could not find a counterexample. Would you help me out?

Answer 1

The functions $s_n(x) = x^n$ are continuous on $[0, 1]$, but the limit function $s(x) = \lim_{n \to \infty} s_n(x) $ is not lower semicontinuous.

Now define the functions $f_n$ such that $s_n = \sum_{k=1}^n f_n$ to get a counterexample.

Generally, if $\sum_{k=1}^n f_k$ decreases pointwise then $f = \sum f_n$ is upper semicontinuous. If we choose the $f_n$ so that $f$ is finite but not continuous, then it can not be lower semicontinuous.