Riemann-Lebesgue equivalence for n-dimensional integration

"Lebesgue's Theorem" states that for any bounded $f:[a,b] \to \mathbb{R}$, $f$ is Riemann Integrable iff $m\{x:f \text{ is not continuous at x }\}=0$, and if so Riemann's integral coincides with Lebesgue's. ($m$ is Lebesgue's measure).

Does there exist a generalization of this theorem for higher dimensions? Can I have a proof or a reference please?

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First note that the Lebesgue-Riemann theorem states that $f:[a,b]\to \mathbb R$ is Riemann integrable if, and only if, $f$ is bounded and is continuous almost everywhere (if a function is not bounded, it can't be Riemann integrable).
A proof of the more general result for multivariable functions can be found in volume 1 of "Real Analysis" by Duistermaat. A better exercise in fact will be to take this wiki proof and adapt it to $\mathbb R^n$.