2011/05/30 by Rodrigo López Pouso, Pouso, Rodrigo López
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Black Holes and Theoretical Physics #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical and Theoretical Analysis #math.CA
paper · pdf · doi:10.48550/arxiv.1105.5938
arxiv created 2011/05/30 · openalex publication_date 2011/05/30 · arxiv updated 2011/05/31 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
We approach the Riemann integral via generalized primitives to give a new proof for a general result on change of variable originally proven by Kestelman and Davies. Our proof is similar to Kestelman's, but we hope readers will find it clearer thanks to the use of a new test for the Riemann integrability (which we introduce in this paper) along with some ingredients from some other more recent proofs available in the literature. We also include a bibliographical review of related results and proofs. Although this paper emphasizes in the change of variable theorem, our contributions to the Riemann integration theory are of independent interest. For instance, we present a very simple proof to the fact that continuous functions are integrable which avoids the use of uniform continuity.