What is an example of a lower semicontinuous function that is not continuous? I couldn't find an example anywhere. Does anyone know such example? Thanks.
A useful exercise would be to look at these two functions:

$$f(x) = \begin{cases} 0, \quad &amp; x&lt;0 \\ 1, &amp; x\ge 0\end{cases}$$

$$g(x) = \begin{cases} 0, \quad &amp; x\le0 \\ 1, &amp; x\gt 0\end{cases}$$

One of these is upper semi-continuous and the other is lower semi-continuous. Clearly they only differ when $x=0$, and it would be useful for you to examine the definition of lower semi-continuous very careful for each one to decide which is which.
$f(x)=[x]$ such that $f(x)=1$  if  $x&lt;1$; and  $f(x)=2$ if $x=1$; and $f(x)=\frac{1}{2}$ if $ x&gt;1$.