I was doing some work, and came across an integral in the form $$\int_1^\infty [x] \phi'(x) dx$$ Assume $[x]\phi(x)$ vanishes at infinity.
Now, when I saw this, i was curious to try integration by parts. So I got $$-\phi(1) - \int_1^\infty [x]' \phi(x) dx$$ From here, I tried to make the quick argument because $[x]'$ is undefined at countably many disconnect points, that I can remove those points from the domain of the integral, and conclude that integral is 0. This seems to be wrong, however. Why is that? Is there any way to solve an integral like this? Or is it simply that, the integral doesn't exist, in that choice of $f$ and $g$ in integration by parts.