ABSTRACT
This chapter discusses one general approach to nondifferentiable functions from the viewpoint of category and measure. It demonstrates that, for a given generalized notion of derivative, the set of real-valued continuous nondifferentiable functions turns out to be sufficiently large. The Fubini theorem is fundamental for all of measure theory. Moreover, this theorem has many applications in real analysis, probability theory, and other fields of mathematics. Also, it is well known that the Kuratowski–Ulam theorem possesses a number of nontrivial applications in general topology and mathematical analysis. The chapter points out that the standard operations used in classical mathematical analysis are of projective type, i.e., these operations are described completely by some projective sets lying in certain Polish topological spaces.
