Double integral of two continuous functions

Question

Let $R$ an elemental rectangle in $\mathbb{R}^{2}$ and $g, f: R \rightarrow \mathbb{R}$ continuous functions. Show that $$ \lim _{d(P) \rightarrow 0} \sum_{i, j} f\left(u_{i j}\right) g\left(v_{i j}\right) A\left(R_{i j}\right)=\iint_{R} f(x, y) g(x, y) d A $$ where $P=\left\{R_{i j}: 1 \leq i \leq n, 1 \leq j \leq m\right\}$ is a partition by elementary rectangles of $ R $ and $ u_ {i j}, v_ {i j} $ any points in $ R_ {i j} $.

My problem is that while the result is widely known and proved for only one continuous function, I can find no proofs concerning two continuous functions. All my approaches have failed since the definition of Riemann integral doesn't apply very well here. Any suggestions as to how to proceed would be very appreciated.

Answer 1

The function $fg$ is a product of two continuous functions, and is therefore continuous. So you can apply the known result.

A famous result of Darboux is that the result holds for any Riemann integrable function, i.e. every Riemann integrable function $f : R \to \mathbb{R}$, $R \subset \mathbb{R}^n$ a rectangle, has the property that Riemann sums with respect to any (rectangular) partitions $P_n$ of $R$ whose maximal diameter of a subrectangle goes to $0$ as $n \to \infty$ converge to $\int f(x) \,dx$ as $n \to \infty$. A proof in the case $n = 1$ is given here: https://mtaylor.web.unc.edu/wp-content/uploads/sites/16915/2018/04/anal1v.pdf as theorem 4.2.4 on page 132.