Cauchy Equation and Equations of Cauchy TypeThe equation \( f(x+y)=f(x)+f(y) \) is called the Cauchy equation. If its domain is \( \mathbb{Q} \), it is wellknown 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 counterexample we can show that the solution to the Cauchy equation in this case doesn\( \prime \)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.

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