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It is known (see, for instance, Wheeden & Zygmund's "Measure and Integral", 2nd edition (CRC Press 2015)) that, given a compact interval $I \subseteq \mathbb{R}$ with non-empty interior, and a function $f\in I\mapsto \mathbb{R}$ that is bounded on $I$, $f$ is Riemann-integrable iff $f$ is continuous Lebesgue-almost-everywhere in $I$. Is there a similar characterization of Riemann-integrability for real-valued functions of two real variables (or, more generally, of several real variables)? For instance, does the following statement hold?

Denote the $\sigma$-algebra of Lebesgue-measurable subsets of $\mathbb{R}^2$ by $\mathcal{M}_2$, and denote the Lebesuge measure on $\mathcal{M}_2$ by $\lambda_2$. Let $I, J\subseteq\mathbb{R}$ be compact intervals with non-empty interiors, and let $f\in I\times J\mapsto \mathbb{R}$ be bounded on $I\times J$. Then $f$ is Riemann-integrable on $I\times J$ iff the set $S$ of discontinuities of $f$ in $I\times J$ satisfies: $S \in \mathcal{M}_2$ and $\lambda_2(S) = 0$.

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Yes, this is true (in $\mathbb{R}^n$ for any $n$). See, for example, Section 11.1.2 in Mathematical Analysis II by Vladimir A. Zorich.

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