Riemann Sum

This is a Povray animation I have done 20 years ago (just rendered again in 4K). It illustrates why the double integral is a limit of Riemann sums if the function is continuous and so having a global modulus of continuity. The heavy fluid is the lower bound, the light fluid the upper bound. If each container box has base length dA=dx² then the height of the error is smaller or equal than M dx where M is a modulus of continuity meaning that |f(x,y)-f(a,b)| is bound by a constant M times the distance d((x,y),(a,b)) between the points (this is the maximal possible slope on the surface if the function is differentiable). The volume error in each box is smaller or equal than dV=dh dA = M dx dx² = M dx³. There are about A/dx² containers if the area of the region is A. The total volume of the error is smaller or equal than M A dx which goes to zero if dx goes to zero.

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