Non-differentiability and Hölder properties of self-affine functions

We consider the class of self-affine functions. Firstly, we characterize all nowhere differentiable self-affine continuous functions. Secondly, given a self-affine continuous function ϕ, we investigate its Hölder properties. We find its best uniform Hölder exponent and when ϕ is C1, we find the best uniform Hölder exponent of ϕ′. Thirdly, we show that the Hölder cut of ϕ takes the same value almost everywhere for the Lebesgue measure. This last result is a consequence of the Borel strong law of large numbers.