Distinct Continuous Maps with All Riemann Sums Equal

Do examples as in the title exist? It depends on how the term Riemann sum is understood. For the standard, left, or right Riemann sums such examples do not exist. However, as we will see, they do exist for the lower and upper Riemann sums. Nevertheless, there are only a few examples of such pairs and they have a very simple structure. In this article, we describe all such pairs among Riemann integrable functions from an interval (Formula presented.) into (Formula presented.). We also show that such pairs have an especially nice format when we restrict our attention to continuous maps. All the arguments presented are elementary. In particular, the part concerning continuous functions is self-contained and presented in a format accessible to good undergraduate students.