Continuous functions with impermeable graphs

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

Automated summary

We construct a Holder continuous function on the unit interval that coincides with every function of total variation smaller than 1 passing through the origin at uncountably many points. We introduce the concept of impermeable graph and provide examples of functions with both permeable and impermeable graphs. We also demonstrate that typical continuous functions have permeable graphs. Another major result is the construction of a continuous function on the unit interval that coincides with every function of total variation smaller than 1 passing through the origin in a set of Hausdorff dimension 1.