RT Journal Article
T1 A roller coaster approach to integration and Peano’s existence theorem
A1 López Pouso, Rodrigo
K1 Primitive
K1 Newton integral
K1 Peano’s existence theorem
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
PB Springer
SN 0011-4642
YR 2023
FD 2023
LK http://hdl.handle.net/10347/30840
UL http://hdl.handle.net/10347/30840
LA eng
NO López Pouso, R. A roller coaster approach to integration and Peano’s existence theorem. Czech Math J (2023). https://doi.org/10.21136/CMJ.2023.0514-22
DS Minerva
RD 23 abr 2026