Open AccessThe general theory of integration.
Author(s) -
William Young,
E. W. Hobson
Publication year - 1904
Publication title -
proceedings of the royal society of london
Language(s) - English
Resource type - Journals
eISSN - 2053-9126
pISSN - 0370-1662
DOI - 10.1098/rspl.1904.0059
Subject(s) - interval (graph theory) , mathematics , limiting , upper and lower bounds , function (biology) , calculus (dental) , value (mathematics) , mathematical analysis , pure mathematics , mathematical economics , combinatorics , statistics , engineering , medicine , mechanical engineering , dentistry , evolutionary biology , biology
The paper begins with a recapitulation of the well-known definitions of integration and of upper and lower integration (intégral par excès, par défant ; oberes, unteres Integral). The theorem on which the Darboux definition of upper (lower) integration is founded is stated and proved in the following form :— Given any small positive quantitye 1 , we can determine a positive quantitye , such that, if the fundamental segment S be divided up in any manner into a finite number of intervals, then, provided only the length of each interval is less thane , the upper summation of any function over these intervals differs by less thane 1 from a definite limiting value (the upper integral).