ABSTRACT
In this chapter, the authors complete a systematic study of the odd properties of real functions. They recall that the study of the odd properties of a function reveals its even symmetric structure. Similarly a greater or lesser degree of control on the behavior of the odd part of a function expresses some characteristic of its even symmetric structure. The authors determine the nature of the set of points at which a function (set) may be exactly symmetric. It is easy to see that a function can have many points of symmetry. The authors investigate first the nature of functions that are symmetrically monotonic at every point and then turn to the study of the set of points of symmetric increase for arbitrary functions. They also turn to the problem of determining the continuity properties of functions that are symmetrically continuous.
