zbMATH — the first resource for mathematics

Functions of several variables of finite variation and their differentiability. (English) Zbl 0886.26007
Let \(f\) be a real-valued function defined on the unit interval of \(\mathbb{R}^2\). It is said that \(f\) is nondecreasing if \(f(x_1,y_1)\leq f(x_2,y_2)\) for any \(x_1\leq x_2\) and \(y_1\leq y_2\). Moreover, it is said that \(f\) is of finite variation if the following functions are of bounded variation: \(x\mapsto f(x,0)\), \(y\mapsto f(0,y)\), \([x,y]\times [u,v]\mapsto f(y,v)- f(y,u)- f(x,v)+ f(x,u)\).
The main results of the paper are: (1) Each nondecreasing function is a.e. differentiable. (2) A function has finite variation if and only if it is the difference of two nondecreasing functions. In particular, each function of finite variation is a.e. differentiable.

26B30 Absolutely continuous real functions of several variables, functions of bounded variation
26B05 Continuity and differentiation questions
Full Text: DOI