ABSTRACT
The study of additive functions dates back to A.M. Legendre who first attempted to determine the solution of the Cauchy functional equation
f(x+ y) = f(x) + f(y)
for all x, y ∈ R. The systematic study of the additive Cauchy functional equation was initiated by A.L. Cauchy in his book Cours d’Analyse in 1821. Additive functions are the solutions of this additive Cauchy functional equation. This chapter gives an account of additive functions. First, we explain what a functional equation is. Then we treat the additive Cauchy functional equation and show that continuous or locally integrable additive functions are linear. We further explore the behavior of nonlinear discontinuous additive functions and show that they display a very strange behavior: their graphs are dense in the plane. To this end, we briefly discuss the Hamel basis and its use for constructing discontinuous additive functions. We also examine under what other criteria the solution of the Cauchy functional equation is linear. A discussion of complex additive functions is also provided in this chapter. The chapter ends with a set of concluding remarks where we have pointed out some developments and some open problems related to the additive Cauchy functional equation.
