ABSTRACT
This chapter attempts a rather more systematic study of just the even properties. It repeats to a small degree material but will also delve much more deeply into these matters. Recall that the even properties of a function reflect in some way the odd symmetric structure of the function. In the presence of a regularity condition for example measurability the two concepts, convexity and midpoint convexity, are equivalent and so in the literature either definition may be found labeled simply as “convex”. The chapter shows that a continuous, smooth function must have many points of differentiability and a continuous, quasi-smooth function may have none. It establishes a relationship that must hold between the ordinary derivative and the approximate derivative for quasi-smooth functions. The differentiability properties of continuous, smooth functions were first explored in Rajchman; he showed that the derivative must exist on a dense set.
