On a class of generalized Takagi functions with linear pathwise quadratic variation

We consider a class XX of continuous functions on [0,1][0,1] that is of interest from two different perspectives. First, it is closely related to sets of functions that have been studied as generalizations of the Takagi function. Second, each function in XX admits a linear pathwise quadratic variation and can thus serve as an integrator in Föllmer's pathwise Itō calculus. We derive several uniform properties of the class XX. For instance, we compute the overall pointwise maximum, the uniform maximal oscillation, and the exact uniform modulus of continuity for all functions in XX. Furthermore, we give an example of a pair x,y∈Xx,y∈X for which the quadratic variation of the sum x+yx+y does not exist.