Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4616131 | Journal of Mathematical Analysis and Applications | 2014 | 18 Pages |
Abstract
We present some new results on monotone metric spaces. We prove that every bounded 1-monotone metric space in Rd has a finite 1-dimensional Hausdorff measure. As a consequence we obtain that each continuous bounded curve in Rd has a finite length if and only if it can be written as a finite sum of 1-monotone continuous bounded curves. Next we construct a continuous function f such that M has a zero Lebesgue measure provided the graph(f|M) is a monotone set in the plane. We finally construct a differentiable function with a monotone graph and unbounded variation.
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Physical Sciences and Engineering
Mathematics
Analysis
Authors
AleÅ¡ Nekvinda, DuÅ¡an Pokorný, Václav Vlasák,