1
$\begingroup$

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?

$\endgroup$
3
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.