# Are the conditions for multivariable integrability the same?

For a single variable function, the function needs to have a finite number of discontinuities and must be bounded over the interval of integration for it to be Riemann integrable over that interval.

Are the conditions the same for the multivariable case?

In other words, is it true that if a multivariable function is bounded over a region and has a finite number of discontinuities it is necessarily Riemann integrable?

• Actually (in the single-variable case) it's enough that the discontinuities form a set of Lebesgue measure zero. They don't have to be finitely many. Sep 30, 2014 at 15:01
• Based on your answers to other questions, I'm tempted to simply write "the conditions might be the same, I'm sure you can find them." Even though we have thousands of users, we're still a fairly small community--being unhelpful in other areas of the site can make people unwilling to help you. Sep 30, 2014 at 15:12

Below is the graph of a two variable bounded function from $$[0,1]\times[0,1]$$ that has discontinuities at the boundary of the unit disk.
If we use a denser subdivision, the orange pillars become thinner, and the green boxes become flatter $$\implies$$ their combined volume $$\to0$$.