I gave my generalisation in terms of frequency, and not in terms of "number of rectangles" in the Riemann sum, which I conveniently assumed to be infinite.

A better generalisation: if a function is thin-tailed, and its Fourier transform is thin-tailed, then it's Riemann sums converge to its integral at an exponential rate. In particular, this applies to the Gaussian function.

I'll check out what you said.