Solve the functional equation $4f(x)=f(2x)$

Question

Solve the functional equation $4f(x)=f(2x)$. As for now I know that one solution is $f(x)=cx^2$, where c is a constant value.

Answer 1

If $f$ is not necessarily continuous, then there are many solutions. It can, for instance, be defined as $$ f(x) = \cases{x^2 & if $x$ is rational\\ 3x^2 & if $x$ is a rational multiple of $\pi$.\\ -5x^2 & if $x$ is a rational multiple of $e$\\ x^2\ln 2 & if $x$ is a rational multiple of $\sqrt{2}$\\ 0 & otherwise} $$ and even the rational numbers (and thus each of the posts above) can be further subdivided into more and more subsets of the real line that will never interfere with each other through the functional equation, and therefore can have their own definition.