Publication at Faculty of Mathematics and Physics |
2014
Abstract
A metric space (X,d) is monotone if there is a linear order 0 such that d(x,y)a parts per thousand broken vertical bar cd(x,z) for all x < y < zaX. Properties of continuous functions with monotone graph (considered as a planar set) are investigated.
It is shown, for example, that such a function can be almost nowhere differentiable, but must be differentiable at a dense set, and that the Hausdorff dimension of the graph of such a function is 1.