Precision needed in definition of unboundedness

Question

This is a quick question about what it means for a function to be unbounded. Does it mean that the function tends to + or - infinity, or does it just mean that it has no limit?

Answer 1

Neither. It means that there does not exist a value $M>0$ such that for all $x$, $f(x)\in [-M,M]$. In other words, the range of the function cannot be bounded. Note that the function CAN still have a limit at infinity that is bounded, as the unbounded behavior can occur someplace beforehand. For example, consider the function $f(x)=\frac 1 x$ on $(0,\infty)$. This is unbounded as the closer you get to 0, the bigger $f(x)$ gets, but the limit exists as $x\to \infty$ and is $0$

Edit to answer the comment: For an unbounded function that still can be integrated, take a piecewise defined function, with the main part being any integrable function (For now we'll use the constant 0, for simplicity) and the "rare" part being that on the natural numbers, $f(n)=n$. (In Lebesgue integration, this would be called a set of measure 0, as it is countable).

$\int _0 ^\infty f(x) dx=0$, as we can create a partition such for any $\epsilon >0$, we surround each natural number with an interval no bigger than $\frac \epsilon {n 2^n}$. Then the contribution of that part of the integral to our Riemann sum is $n \cdot \frac \epsilon {n 2^n}=\frac \epsilon {2^n}$, and adding up all of those over the natural numbers, we get $\epsilon$. (And of course the integral on the other parts 0)

It's unbounded obviously it can be as big as any natural number.

The short summary here is, yes it can be integrable as long as the unbounded behavior is sufficiently "small"