Acta Mathematica Hungarica

, Volume 142, Issue 1, pp 1–30 | Cite as

Properties of functions with monotone graphs

  • Michael Hrušák
  • Tamás Mátrai
  • Aleš Nekvinda
  • Václav Vlasák
  • Ondřej Zindulka


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<zX. 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.

Key words and phrases

monotone metric space continuous function graph derivative approximate derivative absolutely continuous function σ-porous set 

Mathematics Subject Classification

26A24 26A27 26A46 


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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2013

Authors and Affiliations

  • Michael Hrušák
    • 1
  • Tamás Mátrai
    • 2
  • Aleš Nekvinda
    • 3
  • Václav Vlasák
    • 4
  • Ondřej Zindulka
    • 3
  1. 1.Instituto de MatemáticasUNAMMoreliaMéxico
  2. 2.Alfréd Rényi Institute of MathematicsHungarian Academy of SciencesBudapestHungary
  3. 3.Department of Mathematics, Faculty of Civil EngineeringCzech Technical UniversityPrague 6Czech Republic
  4. 4.Department of Mathematical Analysis, Faculty of Mathematics and PhysicsCharles UniversityPrague 8Czech Republic

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