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Cauchy Functional Equation

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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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