Can an unbounded function have a finite integral?

Question

I am wondering whether there exists a function such that:

$$\lim_{x \rightarrow a}f(x)=\infty$$

at some point $a$ on the real axis but yet,

$$\int_{-\infty}^{+\infty}\left|f(x)\right|\ dx<\infty$$

Does the fact that a function is unbounded imply that it has no finite integral?

Answer 1

Consider $f(x) = \dfrac{e^{-x^2}}{|x|^{1/2}}$ or something like that.

Answer 2

The function $f(x)=1/\sqrt{x}$ is unbounded at $x=0$, but it has a finite integral $$\int_0^af(x)dx$$ for any positive finite $a$.