Properties of functions with monotone graphs
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A metric space (X,d) is monotone if there is a linear order < on X and a constant c>0 such that d(x,y)≦cd(x,z) for all x<y<z∈X. 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.
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- Properties of functions with monotone graphs
Acta Mathematica Hungarica
Volume 142, Issue 1 , pp 1-30
- Cover Date
- Print ISSN
- Online ISSN
- Springer Netherlands
- Additional Links
- monotone metric space
- continuous function
- approximate derivative
- absolutely continuous function
- σ-porous set
- Author Affiliations
- 1. Instituto de Matemáticas, UNAM, Apartado Postal 61-3, Xangari, 58089, Morelia, Michoacán, México
- 2. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, H-1053, Budapest, Reáltanoda u. 13–15, Hungary
- 3. Department of Mathematics, Faculty of Civil Engineering, Czech Technical University, Thákurova 7, 16000, Prague 6, Czech Republic
- 4. Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 18675, Prague 8, Czech Republic