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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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