TITLE: What does 'finite-valued' mean?
QUESTION [5 upvotes]: In Rudin, while defining the concept of 'pointwise bounded', it says:
if there exists a finite-valuded function $\phi$  defined on $E$ such that $|f_{n}(x)|<\phi (x)$. Here, I am quite puzzled by the definition of finite-valued. Is $f(x)=1/x$ an example of finite-valued function, since when $x$ tends to zero, it tends to infinity, how could it be classified as finite-valued? And any example of a function which is not finite-valuded? 
Thanks so much for your help!

REPLY [3 votes]: Earlier in Rudin he defines the extended real numbers $R\cup\{\pm \infty\}$. He means that the function is real valued and not extended-real valued.