I am trying to understand the relationship between Darboux integrals and different sets in $\mathbb{R}^n$. Does there exist a bounded function $f:\mathbb{R}^n\to\mathbb{R}$ that is Darboux integrable on a bounded set $S$ and its interior, but not integrable on the boundary and closure of $S$?
I am thinking of something similar to $\mathbb{Q}\cap[0,1]$, but I'm not sure.