Functional equations (Table of contents)

Cauchy Equation and Equations of Cauchy Type

The equation \(f(x+y)=f(x)+f(y)\) is called the Cauchy equation. If its domain is \(\mathbb{Q}\), it is well-known that the solution is given by \(f(x)=xf(1)\). That fact is easy to prove using mathematical induction. The next problem is simply the extention of the domain from \(\mathbb{Q}\) to \(\mathbb{R}\). With a relatively easy counter-example we can show that the solution to the Cauchy equation in this case doesn't have to be \(f(x)=xf(1)\). However there are many additional assumptions that forces the general solution to be of the described form. Namely if a function \(f\) satisfies any of the conditions:

then the general solution to the Cauchy equation \(f:\mathbb{R}\rightarrow S\) has to be \(f(x)=xf(1)\).

The following equations can be easily reduced to the Cauchy equation.