A singular function with a non-zero finite derivative on a dense set with Hausdorff dimension one

This article closes a trilogy on the existence of singular functions with non-zero finite derivatives. In two previous papers, the authors had exhibited a continuous strictly increasing singular function from [0,1] into [0,1] with a derivative that takes non-zero finite values at two different zero-measure sets: first, at the points of an uncountable set; then at the points of a dense set in [0,1]. In the present paper, the possibilities are further stretched as the construction is improved to extend it to an uncountable dense set whose intersection with any interval (a,b) has Hausdorff dimension one. Another feature of this third article is the construction of the required function using the most paradigmatic of the singular functions: the Cantor-Lebesgue one.