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On the derivative with respect to a function with applications to Riemann-Stieltjes integral. (English) Zbl 0792.26003
In order to obtain some properties of the second order differential operators, W. Feller introduced and studied a kind of derivative of a function with respect to another function which is strictly increasing [Commun. Pure Appl. Math. 8, 203-216 (1955; Zbl 0068.097); Ill. J. Math. 1, 459-504 (1957; Zbl 0077.291); 2, 1-18 (1958; Zbl 0078.076)].
In the present paper, the author considers in Feller’s definition of the above mentioned derivative instead of the value of the function in a point their unilateral limits in this point and obtains some properties completely analogous to the classical ones such as: Rolle’s theorem, Cauchy’s theorem or a Leibniz-Newton formula for the Riemann-Stieltjes integral.
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
26A42 Integrals of Riemann, Stieltjes and Lebesgue type