Saturday, December 9, 2017

functions - Solution to functional equation and Cauchy

I want to show that the only continuos solutions to the functional equation $g(x)+g(y)=g(\sqrt{x^2+y^2})$


is $g(x)=cx^2$ where c is a constant i.e. c=g(1).


I think that I maybe could rewrite the equation, such that I could use the solution to Cauchy's functional equation i.e.


$f(x+y)=f(x)+f(y)$ with solutions $cf(x)$, but i can't see how. Can anyone give med a hint?

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