I think this is not a good question.

1) Does the interval have anything to do with it?

2) "THE" Riemann Sum?

Given an interval size, it would not be difficult to construct such an animal. It would not be very smooth, I would think. Change the interval size and you must start the construction from the beginning.

Of course, it is trivial to construct an example where the left, mid, right, min, and max are exactly the same.