Are functions which take on the values $\pm \infty$ considered unbounded? I reading some bits and pieces of analys (some, Sakarchi and Stein, Tao). In general, if $f(0) = +\infty$ and $\forall (x \not  = 0) |f(x)|<10$, would $f$ be called bounded or unbounded?
Maybe more generally, is any function in the extended real numbers considered unbounded? Or do we usually say they are all bounded, because they are bounded by $+ \infty$?
 A: I can think of two definitions of boundedness that could be used here.
Boundedness on ordered sets, where a set is bounded if there exist an element greater than or equal to every element in the set, and same for lesser than. 
This concept is useless on the extended reals, as all sets will be bounded. 
Boundedness on metric spaces, where a set is bounded if it can be contained in a ball. 
The extended reals is not a metric space for the usual metric on the real numbers, but there exists metrics that turn it into a metric space. But then the distance from $-\infty$ to $\infty$ must be some finite number, and the whole extended real line must be contained in a ball of this radius (any sane metric on the extended reals should have these two numbers furthest away from each other).
Again, the concept of boundedness is uninteresting, as all sets are bounded.
In general if someone talked about sets being bounded, I would therefore assume they are restricting themselves to the definition of boundedness on the reals. 
For your example though, if we restrict to the reals the function will not be defined at $0$, so it would be bounded on its domain. So for us to say it was unbounded we would have to specifically define what we mean with unboundedness on the extended reals in such a way that anything containing $\infty$ would be unbounded.
