In this talk we will explore analytical properties, such as Hölder continuity, of functions on [0,1] through the lens of continuum tree spaces. These tree spaces originally came from probability theory and the study of Brownian motion but strongly link the regularity of functions with the dimension theory of its "dual" continuum tree space. In this talk we will show that using tools from dimension theory to those dual spaces is the "right thing" to do. As an application we provide a concise proof of a theorem by Picard that links Hölder regularity with the upper box counting dimension of the tree, along with some new results.

Based on joint work with Maik Gröger.