ABSTRACT
In various questions of mathematical analysis, one often needs to consider some nontrivial restrictions of a given function, which have nice additional descriptive properties. In general, these properties do not hold for the original function but may be valid for its restrictions to certain non-small subsets of its domain. In order to illustrate this circumstance, this chapter recalls two widely known statements from the theory of real functions. The first of them is the classical theorem of Luzin concerning the structure of an arbitrary Lebesgue measurable function from R into R. Undoubtedly, this theorem plays the most fundamental role in real analysis and topological measure theory. The chapter focuses on Blumberg’s theorem and on some strange functions that naturally appear when one tries to generalize the theorem in various directions. It also shows that any Sierpinski–Zygmund function (defined on R) is totally discontinuous with respect to the family of all subsets of R having the cardinality of the continuum.
