Additive Cauchy Equation

The functional equation

$$f(x+y)=f(x)+f(y)$$

is the most famous among the functional equations. Already in 1821, A. L. Cauchy solved it in the class of continuous real-valued functions. It is often called the

additive Cauchy functional equation

in honor of A. L. Cauchy. The properties of this functional equation are frequently applied to the development of theories of other functional equations. Moreover, the properties of the additive Cauchy equation are powerful tools in almost every field of natural and social sciences. In Section 2.1, the behaviors of solutions of the additive functional equation are described. The Hyers–Ulam stability problem of this equation is discussed in Section 2.2, and theorems concerning the Hyers–Ulam–Rassias stability of the equation are proved in Section 2.3. The stability on a restricted domain and its applications are introduced in Section 2.4. The method of invariant means and the fixed point method will be explained briefly in Sections 2.5 and 2.6. In Section 2.7, the composite functional congruences will be surveyed. The stability results for the Pexider equation will be treated in the last section.