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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.

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