Improper integration and boundedness of function

Question

This question came to my mind when I was following the book of Goldberg on Real Analysis for introduction of improper integrals.
Suppose $f:[a,infty)$ is a function such that (i) $f$ is integrable on $[a,b]$ for every $b>a$. (ii) $lim_{brightarrowinfty} int_a^b f(x),dx$ exist.
Does it follow from these two conditions that $f$ is bounded on whole domain $[a,infty)$?
(I am studying integrals of this kind seriously first time, so this question come to my mind. I do not know example; if there is trivial example, one may put in comment, and then vote for close, that is also fine to me.)

0XLR · Accepted Answer

Not necessarily. Let $f : [1, infty) to mathbb{R}$ be defined as:
$$
f(x) = begin{cases} x &text{ if } x in mathbb{N} \
e^{-x} &text{ otherwise}
end{cases}
$$ Then, for any $b geq 1$, $f$ only has a finite number of discontinuities over $[1, b]$; so, of course, $int_{1}^b f(x) dx = int_{1}^b e^{-x} dx$ . Hence, $int_{1}^infty f(x) dx$ exists and is just $int_{1}^infty e^{-x} dx$. However, obviously $f$ is not bounded on $mathbb{N}$.

Improper integration and boundedness of function

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