Continuous families of Hölder functions that are not of bounded variation

This paper deals with three classes of functions of great importance in analysis and its applications. We construct a family of Hölder functions in the closed unit interval having two continuous parameters. Those functions are not of bounded variation for any pair of values of the Hölder constant and exponent. The construction depends on a change of variables given by a Lipschitz function with constant equal to 1. Several questions related to the concepts of genericity, surjectivity and deformability are posed at the end.