Here, $||P||$ is the norm of the partition which is defined by $max${$\Delta x,\Delta y$}, i.e., the maximum of either side of each sub-rectangle. Why can we not define $||P||$ as $max${$\Delta x \Delta y$}? Can someone provide a counter-example of a function explaining why the following assumption as the norm fails in certain cases? I was told that the Darboux Integral failed to hold for certain integrable functions for the above assumed norm. Also, can we not define $||P||$ as the length of the largest diagonal of all such sub-rectangles? Would there be any problem in that case?
Consider $f(x,y) =x$ on the rectangle $[0,1]^2$, where
$$\int_{[0,1]^2} f = \frac{1}{2}$$
Let the partition $P_n$ have subrectangles $[0,1]\times [(j-1)/n,j/n]$ for $j=1,\ldots,n$. The upper Darboux sum $U(P_n,f)$ remains constant with value $1$ even as $n \to \infty$ and does not converge to the integral. However, the norm of the partition defined as the maximum area of subrectangles is $1/n$ and tends to $0$.