Find multiplicative function $f:\Bbb R\to$ $p$th roots of unity

Question

For prime number $p$, find all functions $f:\Bbb R\setminus\{0\}\to\left\{\mathrm e^{\frac{2k\pi\mathrm i}p}\mid k\in\Bbb Z_+\right\}$ such that $f(ab)=f(a)f(b)$ for all real numbers $a$, $b$.

This problem is very hard, even for small $p$. I tried $f(a)=f(1)f(a)$, so $f(1)=1$. Therefore, $$f(a)f\left(\frac1a\right)=f(1)=1,~\forall a\in\Bbb R\setminus\{0\}.$$ I can't derive anything more.