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On mappings of bounded variation. (English) Zbl 0940.26009
Summary: We present properties of mappings of bounded variation defined on a subset of the real line with values in metric and normed spaces and show that major aspects of the theory of real-valued functions of bounded variation remain valid in this case. In particular, we prove the structure theorem and obtain the continuity properties of these mappings as well as jump formulas for the variation. We establish the existence of Lipschitz continuous geodesic paths and prove an analog of the well-known Helly selection principle. For normed space-valued smooth mappings we obtain the usual integral formula for the variation without the completeness assumption on the space of values. As an application of our theory we show that compact set-valued mappings (= multifunctions) of bounded variation admit regular selections of bounded variation.

MSC:
26A45 Functions of bounded variation, generalizations
26E25 Set-valued functions
49J53 Set-valued and variational analysis
54C60 Set-valued maps in general topology
54C65 Selections in general topology
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