What would be an unbounded set $A$ in $\mathbb{R}^3$ so that every constant is integrable on $A$? Find a continuous function $A \rightarrow \mathbb{R}$ that is not integrable on $A$.
I think that $A = \{0\} \times \{0\} \times \mathbb{R}$ would be a set like that. A function that is not integrable on $A$ would be $f(x, y,z) = z$. Would that be correct? If not do you have any suggestions? How would I prove that that function is not integrable?