Topological Hausdorff dimension and level sets of generic continuous functions on fractals

► We examine a new fractal dimension, the so called topological Hausdorff dimension. ► The generic continuous function has a level set of maximal Hausdorff dimension. ► This maximal dimension is the topological Hausdorff dimension minus one. ► Homogeneity implies that “most” level sets are of this dimension. ► We calculate the various dimensions of the graph of the generic function.