Differentiation with respect to a function of limited variation
P. J. Daniell · Transactions of the American Mathematical Society · 1918
Lebesgue, Radon, and Youngf have defined integrals with respect to a function of limited variation, and these are generalizations of the Stieltjes integral.The next step which suggests itself is a definition of the corresponding derivative.Such a definition is given in this paper, and the fundamental property of a derivative is proved by means of a modification of Vitali's theorem.The steps taken are parallels of steps given by de la Vallée Poussin t in the theory of Lebesgue plane sets.Since this paper was first written, a paper by Young § has appeared, giving a slightly different definition of the derivative, and an entirely different trea ment.The applications at the end of this paper are not given by Young.Definition of derived numbers and derivative.Consider two functions of x, F(x), a(x), defined in the fundamental interval, 0 Si x Si 1.The ratio F(x+ ) -F(x-() AF a(x + e) -a(x -e) Aa may have upper and lower limits as e approaches 0. We define these as the upper and lower derived numbers of F ( x ) with resp ct to a ( x ), and we use the notation DaF(x)-m£, D.F(x)-m£.For x equal to 0 or 1 it is necessary to add a convention whereby F ( x ), a ( x ) are continued beyond the range ( 0, 1 ), and have values equal to their values at 0 or 1 respectively.If the two derived numbers are finite and equal,