A parameterized family of continuous functions which was considered by the first author is re-visited in the case when they are monotonically increasing. We prove that the functions are not only continuous and strictly increasing but also singular, i.e., their derivatives are zero almost everywhere.
Hisashi Okamoto. Marcus Wunsch. "A geometric construction of continuous, strictly increasing singular functions." Proc. Japan Acad. Ser. A Math. Sci. 83 (7) 114 - 118, July 2007. https://doi.org/10.3792/pjaa.83.114