Question :  Let $f$ be real valued function on $R^3$ satisfying (for a fixed $\alpha \in \mathbb R$) $f(rx) = r^{\alpha}f(x)$ for any $r&gt;0$ and $x \in \mathbb R^3$. Then which of the following statements are true.   If $f(x) = f(y)$ whenever $||x|| = ||y|| = \beta$ for a $\beta &gt;0$, then $f(x) = \beta ||x||^{\alpha}$ If $f(x) = f(y)$ whenever $||x|| = ||y|| = 1$ for a $\beta &gt;0$, then $f(x) = ||x||^{\alpha}$ If $f(x) = f(y)$ whenever $||x|| = ||y|| = \beta$ for a $\beta &gt;0$, then $f(x) = c||x||^{\alpha}$ for a some constant c. If $f(x) = f(y)$ whenever $||x|| = ||y|| $ , then $f$ must be constant function.   Please help me to see this question more clearly and help me to answer this.
$f:\mathbb{R}^3\to\mathbb{R}, f(rx)=r^{\alpha}f(x)$ for some $\alpha&gt;0$, $\forall x\in\mathbb{R}$, for any $r\in\mathbb{R}$, could any one tell me which of the following is true?  $1.$ If $f(x)=f(y)$ whenever $||x||=||y||=\beta&gt;0$, then $f(x)=\beta ||x||^{\alpha}$  $2.$ If $f(x)=f(y)$ whenever $||x||=||y||=1$, then $f(x)= ||x||^{\alpha}$  $3.$ If $f(x)=f(y)$ whenever $||x||=||y||=1$, then $f(x)=c||x||^{\alpha}$ for some constant $c$  $4.$ If $f(x)=f(y)$ whenever $||x||=||y||$, then $f(x)$ must be a constant function. Thank you.
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