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?


Take $y=x$ to get $2g(x)=g(\sqrt{2}|x|)$ so $g(x)=g(-x)=g(|x|)$.

Take $y=0$ to get $g(0)=0$. It suffices to find this function only for $x\ge0$ because it is even.

Now, note that the function $f(x):=g(\sqrt{x})$, for $x\ge0$ satisfies the Cauchy equation.

  • $\begingroup$ Thank you! But how does the function $f(x):=g(\sqrt{x})$ satisfy the Cauchy equation? $\endgroup$ – mathstudent Dec 12 '14 at 16:25
  • $\begingroup$ Because $f(x^2) + f(y^2) = f(x^2+y^2)$. $\endgroup$ – M. Wind Dec 12 '14 at 16:34

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