Discontinuous additive functions: Regular behavior vs. pathological features

Abstract

This paper deals with properties of discontinuous additive functions (a function f is said to be additive if f(x + y) = f(x) + f(y) for all x and y). We refer to two different frameworks, namely Q[sqrt 2] (where the axiom of choice is not needed) and the whole set R. We construct an additive function which is both periodic and quasiperiodic (in the sense of definition 1), as well as two periodic functions whose sum is the identity function (see also [9]). We establish several properties and characterizations of additive functions: among these, the fact that the graph of an additive function is homogeneous, and that every straight line passing through a point of the graph divides it into two congruent parts. We introduce two metaphors to describe such properties informally. Several other aspects are discussed, such as arcwise connectedness and Lebesgue measurability. Lastly, we examine the consequences of changing the topology on R

Graphs of real functions with pathological behaviors

“…7. Some results -Let me briefly state some theorems about additive functions and everywhere surjective functions (see for instance [3]). Some of them are simple, while others require a not completely obvious proof.…”