Continuous functions with impermeable graphs

We construct a Hölder continuous function on the unit interval which coincides in uncountably (in fact continuum) many points with every of total variation smaller than 1 passing through origin. conclude that this has impermeable graph—one key concepts introduced paper—and we present further examples functions both permeable and graphs. Moreover, show typical sense Baire category) have The first example is subsequently used to an plane intrinsically Lipschitz complement graph graph, but not plane. As another main result, set Hausdorff dimension passes