Linear Functional Equations

Abstract

A function is called a linear function if it is homogeneous as well as additive. The homogeneity of a function, however, is a consequence of additivity if the function is assumed to be continuous. There are a number of (systems of) functional equations which include all the linear functions as their solutions. In this chapter, only a few (systems of) functional equations among them will be introduced. In Section 6.1, the superstability property of the “intuitive” system (6.1) of functional equations $f(x+y)=f(x)+f(y)\ {\rm and}\ f(cx)=cf(x)$ which stands for the linear functions is introduced. The stability problem for the functional equation $f(x+cy)=f(x)+cf(y)$ is proved in the second section and the result is applied to the proof of the Hyers–Ulam stability of the “intuitive” system (6.1). In the final section, stability problems of other systems, which describe linear functions, are discussed.