is possible that an unbounded function be integrable?

Question

I know that a function is integrable if : $$ \int f <\infty $$

but, what this say about $f$ ?

Answer 1

If you are talking about Lebesgue integrability, yes. For example, $f(x)=1/\sqrt{x}$ on $(0,1)$ is integrable. It is also integrable as an improper Riemann integral. But for a function on a segment $[a,b]$ to be Riemann integrable (in the proper sense) it has to be bounded.