, Volume 142, Issue 1, pp 1-30
Date: 26 Oct 2013

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<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.

Some of the work on this project was conducted during O. Zindulka’s sabbatical stay at the Instituto de matemáticas, Unidad Morelia, Universidad Nacional Autonóma de México supported by CONACyT grant no. 125108. M. Hrušák gratefully acknowledges support from PAPIIT grant IN101608 and CONACYT grant 80355. A. Nekvinda was supported by MSM 6840770010 and the grant 201/08/0383 of the Grant Agency of the Czech Republic. V. Vlasák was supported by the grant 22308/B-MAT/MFF of the Grant Agency of the Charles University in Prague and by grant 201/09/0067 of the Grant Agency of the Czech Republic. He is a junior researcher in the University Centre for Mathematical Modelling, Applied Analysis and Computational Mathematics (Math MAC). Sections 1, 2, 3, 4, 6 and 8 are due to Zindulka, except 2.2 (Mátrai), 2.3 (Mátrai and Vlasák) and 4.1 (Nekvinda); Section 5 is due to Mátrai and Vlasák; Section 7 is due to Hrušák and Zindulka.
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