I was confused in lecture today that the professor said a l.s.c function can not blow up to $+\infty$...
In my point of view, a l.s.c function can blow up to $+\infty$... for example, we take $f(x)=1/|x|$. Clearly we have $f(0)=\infty$ at $0$ and I think $f(x)$ is l.s.c since for any $x_n\to 0$, $\liminf f(x_n)=\infty=f(0)$...
In high dimension case, we can construct a similar example...
Am I right? Or I miss sth trivial here?
Thank you!
A lower semicontinuous and convex functions that is $-\infty$ for some $x$ is nowhere finite. Or in other words, if it is finite for some $x_0$ then it is never $-\infty$. Check also proper convex functions.
The indicator function of a closed and convex set is the prime example of a lower-semicontinuous mapping that takes the value $+\infty$ somewhere. (Even in infinite-dimensional spaces)
Maybe somebody mixed up the things?