Find all functions $f(x+y)=f(x^{2}+y^{2})$ for positive $x,y$ Find all functions $f:\mathbb{R}^{+}\to \mathbb{R}$ such that for any $x,y\in \mathbb{R}^{+}$ the following holds:
$$f(x+y)=f(x^{2}+y^{2}).$$

If $x+y=a$ then $x^2+y^2$ is onto the range $\left[\frac{a^2}{2},a^2\right)$. So $f$ must be constant on any interval $\left[b,2b\right)$ with $b\in\mathbb R^+$.
Now define $b_n=2^{n/2}$ for $n\in\mathbb Z$. Then show that $\bigcup_n [b_n,2b_n)=\mathbb R^+$ and that $b_{n+1}\in(b_n,2b_n)$. This means that $f$ must be constant on all of $\mathbb R^+$.
