A smooth finite-variation integrand and its deterministic error bound

Given: The function $f(x,y)=xy$ on the unit square and the point set containing $(1\mathrm{/}4,1\mathrm{/}2)$.

For $C^d$ functions, upper-anchor Hardy–Krause variation is the sum of the integrals of the absolute mixed derivatives on all nonempty upper faces. (Mixed-derivative formula for smooth Hardy–Krause variation)

The unit square has volume $1$; grid cells' volumes sum to that volume, and for the constant-one function every lower and upper Darboux sum equals the sum of the cell volumes, so its integral is $1$. (Axis-parallel rectangles in ${\mathbb{R}}^m$ and their volume, Grid partitions of a rectangle in ${\mathbb{R}}^m$, their cells, refinements and mesh, Lower and upper Darboux sums over a grid partition in ${\mathbb{R}}^m$, The lower and upper Darboux integrals over a nondegenerate rectangle in ${\mathbb{R}}^m$)

A continuous function on the square is Riemann integrable, and its integrable sections have iterated integrals equal to the multiple integral. (Every continuous function on a closed nondegenerate rectangle in ${\mathbb{R}}^m$ is Riemann integrable, Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections)

Star discrepancy is the supremum over $u\in [0,1{]}^d$ of the absolute difference between the fraction of nodes in $[0,u)$ and its volume. (Star discrepancy, digital nets, and digital sequences)

If $G$ is continuous on $[a,b]$, differentiable on $(a,b)$, and its interior derivative has a Riemann-integrable extension, then its integral is $G(b)-G(a)$. (Newton–Leibniz needs only continuity on $[a,b]$, differentiability on $(a,b)$, and a Riemann-integrable extension of the interior derivative)

For a finite nonempty point multiset and an integrand of finite upper-anchor Hardy–Krause variation, $\mathrm{\mid }{Q}_P(f)-I_d(f)\mathrm{\mid }\le D_N^{\ast }(P)V_{HK,1}(f)$. (Koksma–Hlawka integration bound with upper-anchor variation)