I am looking for all real-valued continuous functions f, on R, which satisfy
$$ f(x)*f(y) = f(x_1)f(y_1) $$ for all x,y, $x_1$, $y_1$, such that $$x^2 + y^2 = (x_1)^2 +(y_1)^2.$$
I don't have much idea on how to solve this problem. The only thing that comes to mind, which doesn't help very much, is the fact that if I let g(x,y) = f(x)*f(y), then since the function g factorizes into two functions of a single variable, we have that the integral of g is the product of the single-variable integrals of f(x)dx and f(y)dy.
Edit: This is not a homework problem. It is a problem that dates back to 2007, as far as I know, and there is a not-so-good solution to it that basically says "guess that the function is Guassian and let's force it to be Guassian." I am looking for another solution to this problem. Thanks.