The upper semicontinuous envelope of a lower semicontinuous function

Question

We call $u^{*}$ is the upper semicontinuous envelope of $u$ if it is the smallest upper semicontinuous function satisfying $u\le u^*$.

My question is that is there any good properties of the upper semicontinuous envelope of a lower semicontinuous function.

Thank you very much!

Answer 1

I want to figure out, in what kind of sense is the upper semi-continuous envelope discontinuous. But that we ask for the function to be lower semi-continuous doesn't play an important role. And the discontinuous points can be dense.