A Remark on the Change of Variable Theorem for the Riemann Integral
In 1961, Kestelman first proved the change in the variable theorem for the Riemann integral in its modern form. In 1970, Preiss and Uher supplemented his result with the inverse statement. Later, in a number of papers (Sarkhel, Výborný, Puoso, Tandra, and Torchinsky), the alternative proofs of these...
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MDPI AG
2021-08-01
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| Online Access: | https://www.mdpi.com/2227-7390/9/16/1899 |
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| author | Alexander Kuleshov |
| author_facet | Alexander Kuleshov |
| author_sort | Alexander Kuleshov |
| collection | DOAJ |
| description | In 1961, Kestelman first proved the change in the variable theorem for the Riemann integral in its modern form. In 1970, Preiss and Uher supplemented his result with the inverse statement. Later, in a number of papers (Sarkhel, Výborný, Puoso, Tandra, and Torchinsky), the alternative proofs of these theorems were given within the same formulations. In this note, we show that one of the restrictions (namely, the boundedness of the function <i>f</i> on its entire domain) can be omitted while the change of variable formula still holds. |
| first_indexed | 2024-03-10T08:37:31Z |
| format | Article |
| id | doaj.art-dce97c1682f742dfb3ca17cd387fa667 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T08:37:31Z |
| publishDate | 2021-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-dce97c1682f742dfb3ca17cd387fa6672023-11-22T08:33:35ZengMDPI AGMathematics2227-73902021-08-01916189910.3390/math9161899A Remark on the Change of Variable Theorem for the Riemann IntegralAlexander Kuleshov0Department of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, RussiaIn 1961, Kestelman first proved the change in the variable theorem for the Riemann integral in its modern form. In 1970, Preiss and Uher supplemented his result with the inverse statement. Later, in a number of papers (Sarkhel, Výborný, Puoso, Tandra, and Torchinsky), the alternative proofs of these theorems were given within the same formulations. In this note, we show that one of the restrictions (namely, the boundedness of the function <i>f</i> on its entire domain) can be omitted while the change of variable formula still holds.https://www.mdpi.com/2227-7390/9/16/1899real analysisRiemann integralchange of variable |
| spellingShingle | Alexander Kuleshov A Remark on the Change of Variable Theorem for the Riemann Integral Mathematics real analysis Riemann integral change of variable |
| title | A Remark on the Change of Variable Theorem for the Riemann Integral |
| title_full | A Remark on the Change of Variable Theorem for the Riemann Integral |
| title_fullStr | A Remark on the Change of Variable Theorem for the Riemann Integral |
| title_full_unstemmed | A Remark on the Change of Variable Theorem for the Riemann Integral |
| title_short | A Remark on the Change of Variable Theorem for the Riemann Integral |
| title_sort | remark on the change of variable theorem for the riemann integral |
| topic | real analysis Riemann integral change of variable |
| url | https://www.mdpi.com/2227-7390/9/16/1899 |
| work_keys_str_mv | AT alexanderkuleshov aremarkonthechangeofvariabletheoremfortheriemannintegral AT alexanderkuleshov remarkonthechangeofvariabletheoremfortheriemannintegral |