Some results on monotone metric spaces

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 | M ) is a monotone set in the plane. We finally construct a differentiable function with a monotone graph and unbounded variation.