- Problem Statement
- Can Riemann sums be non-existant for a discontinuous function on a given interval?

- Relevant Equations
- The definition of Riemann sums (see link below)

The definition of the Riemann sums: https://en.wikipedia.org/wiki/Riemann_sum

I'm stuck with a problem in my textbook involving upper and lower Riemann sums. The first question in the problem asks whether, given a function ##f## defined on ##[a,b]##, the upper and lower Riemann sums for ##f## "always are Riemann sums"? The follow up question asks; if ##f## not only is defined but also continuous on ##[a,b]##, are then the upper and lower Riemann Sums "always Riemann sums"?

The answer to the first question is no and the second yes. Can it be impossible to construct upper and lower Riemann sums for a discontinuous function on a given interval? What is an example of such a function?

I'm stuck with a problem in my textbook involving upper and lower Riemann sums. The first question in the problem asks whether, given a function ##f## defined on ##[a,b]##, the upper and lower Riemann sums for ##f## "always are Riemann sums"? The follow up question asks; if ##f## not only is defined but also continuous on ##[a,b]##, are then the upper and lower Riemann Sums "always Riemann sums"?

The answer to the first question is no and the second yes. Can it be impossible to construct upper and lower Riemann sums for a discontinuous function on a given interval? What is an example of such a function?