Which of the statements are true (CSIR)? 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 $r>0$ 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 $\beta >0$, then $f(x) = \beta ||x||^{\alpha}$

*If $f(x) = f(y)$ whenever $||x|| = ||y|| = 1$ for a $\beta >0$, then $f(x) = ||x||^{\alpha}$

*If $f(x) = f(y)$ whenever $||x|| = ||y|| = \beta$ for a $\beta >0$, 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.
 A: Notice that for $x\neq0$ we have $f(x)=f(\frac{\lvert\lvert x\rvert\rvert}{\beta}\frac{\beta x}{\lvert\lvert x\rvert\rvert})=f(\frac{\beta x}{\lvert\lvert x\rvert\rvert})\beta^{-\alpha}\lvert\lvert x\rvert\rvert^{\alpha}=C\lvert\lvert x\rvert\rvert^{\alpha}$ where $C$ is a constant independent of choice of $x$ since the term on the inside is on the $\beta$ ball about $0$ and by assumption $f(x)=f(y)$ when $\lvert\lvert x\rvert\rvert=\lvert\lvert y\rvert\rvert=\beta$. Thus, 3 is true (notice that if $f(0)=a$ then $a=f(0)=f(r\times0)=r^{\alpha}a$ so either $a=0$ or $r=1$ but this equation is true with arbitrary $r$. Hence, $f(0)=0$ and agrees with $C\lvert\lvert x\rvert\rvert^{\alpha}$ for all $x$.).
We can't necessarily guarantee 1 without more information. For a counterexample to 1 consider $f(x)=-\lvert\lvert x\rvert\rvert^{\alpha}$.
A counterexample to 4 is $f(x)=\lvert\lvert x\rvert\rvert^{\alpha}$. So we can't guarantee 4 either. I'm not sure how to answer 2 since I'm not entirely certain how $\beta$ is involved.
