A century of Sierpinski-Zygmund functions

Sierpinski-Zygmund (SZ) functions are the maps from R to R that have as little continuity as possible. In this work we discuss the history behind their discovery, their constructions through the years, and their generalizations. The presentation emphasizes the algebraic properties of SZ maps and their relation to different classes of generalized continuous-like functions. From the seminal work of Blumberg and the appearance of Sierpinski-Zygmund's result, we describe the current state of the art of this century-old class of functions and discuss the impact that it has had on several different directions of research. Many typical proofs used in the theory, often in a simplified format never published before, are included in the presented material. Moreover, open problems and new directions of research are indicated.