We will recall the definition of semicontinuity of a function and some of related properties. (pdf)
Throughout this note, we will assume that is a metric space and is a Borel measurable function. We say that
- is upper semicontinuous (USC) at , if
- is lower semicontinuous (LSC) at , if is USC at .
- is USC if is USC everywhere, and is LSC if it is LSC everywhere.
It is equivalent to
is USC at if and only if
is USC and increasing if and only if it is RCLL.
— QED —
It’s well known that composition of two continuous functions is continuous. However, it’s not the case for semicontinuity as shown in the next example.
It is false to say “A composition of two USC functions is USC again.” as shown below.
Let be a closed set in .
- The indicator function is USC.
- Let . Then is LSC.
Now, we can generalize the above example.
- is USC at ;
- is an RCLL increasing function,
then is USC at .
Proof: Let’s denote . Fix arbitrary , and we look for , such that
Since is RCLL and increasing, there exists such that , or equivalently
if , then .
On the other hand, since is USC at , there exists such that for all . Thus, the above fulfills the requirement.