The well known [Lebesgue's condition of Riemann integrability](http://en.wikipedia.org/wiki/Riemann_integral) says that a bounded function in one variable
$f\colon [a,b] \to \mathbb{R}$ is Riemann integrable if and only if it is continuous almost everywhere.


I vaguely remember that the same should be true for functions in several variables, say for functions defined on a cube in $\mathbb{R}^n$. 


Is that correct? A reference will be very helpful.



Lebesgue's integrability condition in several variables
This fact is found in T. Apostol, *Mathematical Analysis*, section 10-5. 


The proof is straightforward: if the set of points of discontinuity has positive measure, then the upper and lower sums are far apart. I wanted to include a reference for this in something I was writing (*Classics on Fractals*), but had to ask a lot of people before I found a proof in a textbook.


