I would like to solve the following algebraic linear q-difference equation:

\begin{equation}
a\left(x\right)f\left(x\right)=f\left(qx\right)
\end{equation}

The parameter $q$ is real, positive and smaller than $1$, while the functions $a\left(x\right)$ and $f\left(x\right)$ are $\mathbb{R}\rightarrow\mathbb{R}$. Moreover $a\left(x\right)$ is known and such that $a\left(0\right)=0$ (this is an important constraint), and I would like to calculate $f\left(x\right)$. So for example we can consider the functional equation $xf\left(x\right)=f\left(qx\right)$.

Do you know how to calculate $f\left(x\right)$, given that $f\left(x\right)=0$ for all $x\leq0$ and discontinuous in $x=0$?

Thanks in advance for your help!