Difference between revisions of "Cauchy Functional Equation"

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Revision as of 06:06, 29 March 2008

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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