Article ID Journal Published Year Pages File Type
4616131 Journal of Mathematical Analysis and Applications 2014 18 Pages PDF
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.
Related Topics
Physical Sciences and Engineering Mathematics Analysis
Authors
, , ,