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 $\mathbb R^+$.
For any positives $a,b$ we have:
\begin{aligned} f(a) &amp;=f\left(\frac{a+b}{2}+\frac{a-b}{2}\right) \\
 &amp;=f\left(\big(\frac{a+b}{2}\big)^2+\big(\frac{a-b}{2}\big)^2\right) \\
 &amp;=f\left(\big(\frac{a+b}{2}\big)^2+\big(\frac{b-a}{2}\big)^2\right) \\
 &amp;=f\left(\frac{a+b}{2}+\frac{b-a}{2}\right) \\
 &amp;=f(b)\end{aligned}