Abstract
We present some new results on monotone metric spaces. We prove that every bounded 1-monotone metric space in R-d has a finite 1-dimensional Hausdorff measure.
As a consequence we obtain that each continuous bounded curve in R-d 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 vertical bar M) is a monotone set in the plane.
We finally construct a differentiable function with a monotone graph and unbounded variation. (C) 2013 Elsevier Inc. All rights reserved.