Singular function

Summary

A singular function is a continuous, non-constant function on an interval whose derivative exists and equals zero almost everywhere. Such a function cannot be recovered by integrating its derivative. So it shows that the fundamental theorem of calculus needs absolute continuity in the Lebesgue setting. The Cantor function and the Minkowski question-mark function are the standard examples. Every function of bounded variation is a sum of three parts. They are an absolutely continuous function, a jump function and a continuous function with derivative $0$ almost everywhere.