Discontinuous additive functions: Regular behavior vs. pathological features

作者：Bernardi Claudio^*

来源：Expositiones Mathematicae, 2015, 33(3): 295-307.

DOI：10.1016/j.exmath.2014.10.003

• Periodic function
• Hamel basis
• Graph of an additive function

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). Section 1 has an introductory character: we consider two particular functions, whose domain is Q[root 2] and that can be explicitly defined in an easy constructive way. We show that the considered functions are both periodic and quasiperiodic (in the sense of Definition 1); and we also find that the identity function can be obtained as the sum of two periodic functions (see also Mortola and Peirone (1999)). Then, applying the Axiom of Choice, we consider discontinuous additive functions having the whole set R as domain. 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. Finally, we examine the consequences of changing the topology on R.

• 出版日期2015