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On natural functions and Lipschitz functions. (English) Zbl 1063.26005

Let \(E\subset \mathbb R\) be a nonempty bounded set, let \(X\) be a metric space with metric \(d\). The total variation \(V(f,E)\) of a map \(f\colon~E\rightarrow X\) on \(E\) is defined as \[ V(f,E)=\sup~\left\{\sum_{i=1}^{m}d(f(t_i),f(t_{i-1})): t_0<\dots<t_m,~t_i\in E,\;m\in\mathbb N\right\}. \] A map \(g: E\rightarrow\mathbb R\) is called natural if \(V(g,E\cap [a,b])=b-a\) for all \(a,b\in E\), \(a\leq b\). The author gives certain characterizations of natural functions. He shows that the set of natural functions from \([a,b]\) into a normed space \(X\) is small in the porosity sense in the set of uniform Lipschitz \(1\) functions with the norm inherited from the space of \(BV\) functions. The sizes of other classes of Lipschitz functions in \(BV\) are also established.

MSC:

26A45 Functions of bounded variation, generalizations
26A16 Lipschitz (Hölder) classes
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