$A$-integrability of functions

We give an example of a function which is $A$-integrable on a segment $[a,b]$ and is not $A$-integrable on all subsegments $[a',b']\subset[a,b]$, $[a',b']\ne[a,b]$, $a'\ne b'$. We prove the following theorem. The class of sets $\Bigl\{x\in[a,b]:(A)\displaystyle\int_{x_0}^x f(t)\,dt\,\text{exists}\Bigr\}$, $a\leq x_0\leq b$, is exactly the class of sets which contain $x_0$ and are of the type $F_{\sigma\delta}$ on $[a,b]$.