https://doi.org/10.65770/XKMP8410
ABSTRACT
Differentiation of a function f(x) with respect to a function g(x) is known as the ratio having numerator as the derivative of fwith respect to x and having denominator as the derivative of g with respect to x. This definition is modified as the limit of the ratio ((f(x)-f(s)/(g(x)-g(s)) when g(x) tends to g(s) in a meaningful way. Integration of a function with respect a function of bounded variation is known as a Riemann-Stieltjes integral. The same approach meant for Riemann-Stieltjes integrals is extended for more functions general than functions of bounded variation. These extensions are achieved through a classification of functions which are well defined with respect a given function. More precisely, a function f is said to be well defined with respect to g, when these functions have a common domain, if f(x)=f(y) whenever g(x)=g(y). Continuity of a function with respect to a function is also discussed. It is observed that this approach provides an approach that is easier than the existing ones. It is further observed that the definition of functions well defined with respect to a given function can be extended to a definition of functions well defined with respect to given two functions, and thereby the possibility for extending differentiation to partial differentiation and for extending integration to joint integration with respect to several given functions becomes positive.
References
- [1] W. Botsko, An elementary proof of Lebesgue’s differentiation theorem. The American Mathematical Monthly110(9) (2003) 834-838.
- [2] Browne, and J.D. Cook, An Intuitive Approach to L’Hospital’s Rule. The Mathematics Teacher75(9) (1982) 757-758.
- [3] M. Bruckner, creating differentiability and destroying derivatives. The American Mathematical Monthly85(7) (1978) 554-562.
- [4] S. Carter, L’hospital’s rule for complex-valued functions. The American Mathematical Monthly65(4) (1958) 264-266.
- [5] J. Daniell, Differentiation with respect to a function of limited variation. Transactions of the American Mathematical Society19(4) (1918) 353-362.
- [6] J. Daniell, A general form of integral. Annals of mathematics19(4) (1918) 279-294.
- [7] De Guzman, A change-of-variables formula without continuity. The American Mathematical Monthly87(9) (1980) 736-739.
- [8] Dierolf, and V. Schmidt, A proof of the change of variable formula for d-dimensional integrals. The American mathematical monthly105(7) (1998) 654-656.
- [9] Eisenberg, and R. Sullivan, The fundamental theorem of calculus in two dimensions. The American mathematical monthly109(9) (2002) 806-817.
- Flanders, Differentiation under the integral sign. The American Mathematical Monthly80(6) (1973) 615-627.
- M. Flett, A mean value theorem. The Mathematical Gazette42(339) (1958) 38-39.
- Furi, and M. Martelli, On the mean value theorem, inequality, and inclusion. The American Mathematical Monthly98(9) (1991) 840-846.
- B. Goodner, Mean value theorems for functions with finite derivates. The American Mathematical Monthly67(9) (1960) 852-855.
Download all article in PDF
