examples of the lower semicontinuous functions

Question

The following is the definition of lower semicontinuity:

Let $T$ be a topological space and $f:T\to\Bbb{R}\cup\{+\infty\}$ a function. $f$ is said to be lower semicontinuous if the set $\{x\in T:f(x)>a\}$ is open in $T$ for every $a\in\Bbb{R}$.

The Wikipedia article gives several examples. But I don't see why one would need the extanded value and how it would be useful. Could anyone come up with an example of lower semicontinuous function with the extended value $+\infty$ being assumed at some point in the domain? Why is the extended value useful in this definition?

Answer 1

Let $f$ be continuous and $K \subset T$ a closed subset, then

$$ f_K(x)= \begin{cases} f(x) & \text{if } x\in K \\ +\infty & \text{if } x\notin K \end{cases}$$

is a lower semicontinuous function as

$$\{ f_K >a\} = T\setminus K \cup \{ f>a\}.$$

Answer 2

Lower semicontinuity is also the "right minimum regularity" for superharmonic functions, a good example would be (in the complex plane) $$ u(z) = \log \frac{1}{|z|}. $$ where the value at $0$ is taken as $+\infty$. Now, this example is still continuous (as an extended real-valued function), but if we put $$ u(z) = \sum_{k=1}^\infty \alpha_j \log\frac{1}{|z-\frac1j|} $$ where $\alpha_j$ is small enough to make $u(0) < \infty$, we get something a little more interesting: a lower semi-continuous function where $u(1/j) = +\infty$ for all positive integers $j$, but $u(0) < +\infty$.