What are some examples of upper semicontinuous set valued functions?

Question

I know the definition of an upper semicontinuous set valued function;

A function $f:X\rightarrow 2^Y$ is upper semicontinuous at a point $x \in X$ provided that if $V$ is an open set in Y containing $f(x)$ then there exists an open set $U$ of $X$ that contains $x$ such that if $t \in U$ then $f(t)\subset V$. The function $f$ is upper semicontinuous if $f$ is upper semiconintuous at all $x \in X$

I can use this definition for some proofs in my study of inverse limits however I have no intuitive idea of what this actually means. What are some functions (preferably in the form $f:[0,1]\rightarrow 2^{[0,1]}$) that are and are not upper semicontinuous and what makes them so?

Answer 1

Let $X,Y$ be topological spaces and $f:X\to 2^Y$ a mutlivalued function. Define the graph of $f$:

$$Gr(f)=\big\{(x,y)\subseteq X\times Y\ |\ y\in f(x)\big\}$$

With that the following is true:

Lemma. Assume that $f(x)$ is nonempty and closed for each $x\in X$. If $f$ is upper hemicontinuous then $Gr(f)$ is closed in $X\times Y$. If additionally $Y$ is compact then the converse holds as well: if $Gr(f)$ is a closed subset of $X\times Y$ then $f$ is upper hemicontinuous.

So in case $X=Y=[0,1]$ and each $f(x)$ is nonempty and closed then upper hemicontiunity is simply equivalent to $Gr(f)$ being closed. I'm pretty sure that with that you can find lots of examples and counterexamples.

One such counterexample would be:

$$f(x)=\begin{cases} \{0\} & x\in[0,\frac{1}{2}) \\ \{1\} & x\in[\frac{1}{2}, 1) \end{cases}$$

a counterexample pretty much copied from single-valued case. The graph is not closed, every value is closed, hence the function is not upper hemicontinuous.