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

Proposition 1

is USC at if and only if

–QED–

Example 1

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.

Example 2

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.

Example 3

If

- 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.