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Distinct continuous maps with all Riemann sums equal. (English) Zbl 1452.26010

The paper deals with the following interesting question: Is it possible to find two distinct functions \(f,g:[a,b]\to\mathbb R\) for which all Riemann sums coincide?
The answer depends on the type of the Riemann sums. If the right (or left) Riemann sums of two Riemann integrable functions \(f\), \(g\) coincide, then \(f=g\). However, the same conclusion no longer holds for lower (or upper) Riemann sums. For example, if we take \(m\ne 0\) and consider the distinct functions \(f(x)=mx+\beta\), \(g(x)=-mx+\gamma\), where the parameters \(\beta\), \(\gamma\) are chosen in such a way that \(f(a)=g(b)\), then the lower Riemann sums of \(f\) and \(g\) coincide. The first main result of the paper says that if we restrict ourselves to the classes of continuous functions, then there are no other pairs \(f\), \(g\) whose lower Riemann sums coincide. The proof uses only elementary analysis.
The same problem in the class of all Riemann integrable functions is harder. Given the above-mentioned pair \(f\), \(g\), let \(\tilde f\), \(\tilde g\) be such that \(f\le\tilde f\), \(g\le\tilde g\), \(\tilde f=f\) a.e., and \(\tilde g=g\) a.e. Then the lower Riemann sums of \(f\) and \(g\) coincide. It turns out that this construction gives precisely all pairs of Riemann integrable functions that are not equal a.e., and whose lower Riemann sums coincide.

MSC:

26A42 Integrals of Riemann, Stieltjes and Lebesgue type
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