A roller coaster approach to integration and Peano's existence theorem

Rodrigo López Pouso

Czechoslovak Mathematical Journal (2025)

  • Issue: 1, page 157-177
  • ISSN: 0011-4642

Abstract

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This is a didactic proposal on how to introduce the Newton integral in just three or four sessions in elementary courses. Our motivation for this paper were Talvila's work on the continuous primitive integral and Koliha's general approach to the Newton integral. We introduce it independently of any other integration theory, so some basic results require somewhat nonstandard proofs. As an instance, showing that continuous functions on compact intervals are Newton integrable (or, equivalently, that they have primitives) cannot lean on indefinite Riemann integrals. Remarkably, there is a very old proof (without integrals) of a more general result, and it is precisely that of Peano's existence theorem for continuous nonlinear ODEs, published in 1886. Some elements in Peano's original proof lack rigor, and that is why his proof has been criticized and revised several times. However, modern proofs are based on integration and do not use Peano's original ideas. In this note we provide an updated correct version of Peano's original proof, which obviously contains the proof that continuous functions have primitives, and it is also worthy of remark because it does not use the Ascoli-Arzelà theorem, uniform continuity, or any integration theory.

How to cite

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López Pouso, Rodrigo. "A roller coaster approach to integration and Peano's existence theorem." Czechoslovak Mathematical Journal (2025): 157-177. <http://eudml.org/doc/299928>.

@article{LópezPouso2025,
abstract = {This is a didactic proposal on how to introduce the Newton integral in just three or four sessions in elementary courses. Our motivation for this paper were Talvila's work on the continuous primitive integral and Koliha's general approach to the Newton integral. We introduce it independently of any other integration theory, so some basic results require somewhat nonstandard proofs. As an instance, showing that continuous functions on compact intervals are Newton integrable (or, equivalently, that they have primitives) cannot lean on indefinite Riemann integrals. Remarkably, there is a very old proof (without integrals) of a more general result, and it is precisely that of Peano's existence theorem for continuous nonlinear ODEs, published in 1886. Some elements in Peano's original proof lack rigor, and that is why his proof has been criticized and revised several times. However, modern proofs are based on integration and do not use Peano's original ideas. In this note we provide an updated correct version of Peano's original proof, which obviously contains the proof that continuous functions have primitives, and it is also worthy of remark because it does not use the Ascoli-Arzelà theorem, uniform continuity, or any integration theory.},
author = {López Pouso, Rodrigo},
journal = {Czechoslovak Mathematical Journal},
language = {eng},
number = {1},
pages = {157-177},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A roller coaster approach to integration and Peano's existence theorem},
url = {http://eudml.org/doc/299928},
year = {2025},
}

TY - JOUR
AU - López Pouso, Rodrigo
TI - A roller coaster approach to integration and Peano's existence theorem
JO - Czechoslovak Mathematical Journal
PY - 2025
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
IS - 1
SP - 157
EP - 177
AB - This is a didactic proposal on how to introduce the Newton integral in just three or four sessions in elementary courses. Our motivation for this paper were Talvila's work on the continuous primitive integral and Koliha's general approach to the Newton integral. We introduce it independently of any other integration theory, so some basic results require somewhat nonstandard proofs. As an instance, showing that continuous functions on compact intervals are Newton integrable (or, equivalently, that they have primitives) cannot lean on indefinite Riemann integrals. Remarkably, there is a very old proof (without integrals) of a more general result, and it is precisely that of Peano's existence theorem for continuous nonlinear ODEs, published in 1886. Some elements in Peano's original proof lack rigor, and that is why his proof has been criticized and revised several times. However, modern proofs are based on integration and do not use Peano's original ideas. In this note we provide an updated correct version of Peano's original proof, which obviously contains the proof that continuous functions have primitives, and it is also worthy of remark because it does not use the Ascoli-Arzelà theorem, uniform continuity, or any integration theory.
LA - eng
UR - http://eudml.org/doc/299928
ER -

References

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