A Note on the Non-Differentiable Function of Weierstrass

Abstract

has originally been proven t in order to make available, in the case of aii infinite t-range, an analogue to the measure function of Lebesgue. The latter is obtained by a non-local inversion t = t(x) of x = x(t) and has in the classical case of a finite t-range the object of smoothing the behavior of the original function x(t). In fact, the superiority of the Lebesgue integration theory is due, at least in part, to this inversion. It was, therefore, to be expected that to the rather intricate behavior ? of an almost-periodic curve x = x(t) there might correspond an essentially smoother behavior of its distribution function. A proper example has, however, been missing so far.