Cauchy Functional Equation

The Cauchy Functional Equation refers to the functional equation $f:A\to B$ , with $f(x+y) = f(x) + f(y) ,$ for all $x,y \in A$ .

Rational Case

If $A=B=\mathbb Q$ (or any subset closed to addition like $\mathbb Z$ or $\mathbb N$ ), the solutions are only the functions $f(x)=ax$ , with $a\in\mathbb Q$ .

Real Case

If $A=B=\mathbb R$ , then we need a suplementar condition like $f$ continous, or $f$ monotonic, or $f(x)>0$ for all $x>0$ , to get that all the solutions are of the form $f(x)=ax$ , with $a\in\mathbb R$ .

There have been given examples of real functions that fulfill the Cauchy Functional Equation, but are not linear, which use advanced knowledge of real analysis.

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Cauchy Functional Equation

Rational Case

Real Case