Is it true that for a sequence of functions $f_n \in C(K,\mathbb{R})$, where $K \subset \mathbb{R}^n$ the limit is semicontinuous?
I would say that if it's a monotone decreasing lsequence, then the function is lower semicontinuous and otherwise upper semicontinuous. Is this true?
I was just wondering whether my assumption is correct, although, I would try to prove it then by my own.(I am asking so that I don't waste hours on trying to prove something that is not even true)
Adding: Today, our tutor in university told us that it would be possible to construct a sequence of functions $f_n$ such that the limit would be something like this:
This one is clearly NOT lower or upper semicontinuous. Are you still sure about our proof? I have some doubts about it, cause maybe it is problematic that $f_n$ is only continuous on a compact set. This is differenct from being continuous and having compact support. Maybe it does not follow from $f_n \in C(K,\mathbb{R})$, that $f_n$ is semicontinuous everywhere? Though, it could be true that my tutor is wrong. Or do you see a way to approximate the function I have drawn by $f_n \in C(K,\mathbb{R})$ functions?
Best Answer
You have confused your cases. Let's say more generally,
(Depending on the definitions, you may need to assume that the supremum/infimum are finite everywhere.)
For a monotonic sequence, the limit is the supremum/infimum, so that case is subsumed.