If $f(rx)=r^{\alpha} f(x)$, then which of these options is true?

Question

I am trying questions of previous year real analysis quiz and I was unable to solve this particular question .

Let $f$ be a real valued function on $\mathbb R^{3}$ satisfying (for a fixed $\alpha$ belonging to $\mathbb{R}$) , $f(rx)=r^{\alpha} f(x)$ for any $r>0$ and $x$ belonging to $\mathbb R^{3}$. Then which one of the following is true .

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

B. If $f(x)=f(y)$ whenever $\|x\|=\|y\|=1$, then $f(x)=\|x\|^{\alpha}$.

C. If $f(x)=f(y)$ whenever $\|x\|=\|y\|=1$ , then $f(x)=c \|x\|^{\alpha}$ , for some constant $c$.

D. If $f(x)=f(y)$ whenever $\|x\|=\|y\|=1$ , then $f$ must be a constant function .

I tried by assuming $x,y$ to be equal to $\beta$ $e^{i\theta}$ and $\beta$ $e^{i\phi}$ and then equating to find conditions but how to find value of $f(x)$ explicitly using $f(rx)=r^{\alpha}f(x)$.

Kindly guide.

Answer 1

Let's call $c$ the value of $f(x)$ for every $x$ such that $\|x\|=1$.
You can write, for every $x\neq0$, $$ f(x)=f\left(\|x\|\frac{x}{\|x\|}\right)=\|x\|^\alpha f\left(\frac{x}{\|x\|}\right)=(*) $$ and given that $$ \left\|\frac{x}{\|x\|}\right\|=\frac{1}{\|x\|}\|x\|=1 $$ then $$ (*)=\|x\|^\alpha c $$

Answer 2

Your method strictly speaking doesn't work because $\beta e^{i\theta} \in \mathbb C\cong \mathbb R^{\Huge 2}$ but the question is about $\mathbb R^{\Huge 3}$. But the result is true for every $\mathbb R^n$ so it's kind of OK.

In $\mathbb R^2$, your method gives $$ f(\beta e^{i\theta}) =\beta^\alpha f(e^{i\theta})= \beta^\alpha C$$ where $f(e^{i\theta}) = C$ is some constant. (This is because the function $\theta\mapsto f(e^{i\theta})$ is constant; call this constant $C$.) So option C is the correct one, at least for $\mathbb R^2$.

Try and see if you can adapt this proof for $\mathbb R^3$.