Functional Equations and information theory: $f(xy)=f(x)f(y)$ for all $x,y\in[0,1]$

Question

What is the solution of the functional equation $f(xy)=f(x)f(y)$ for all $x$ and $y$ in $I$ where $f$ is a real-valued mapping with domain the unit closed interval $I$?

Answer 1

It is not full solution but it can help you solve it fully (if I solve it fully I will edit this). Take $y=x$, then $f(x^2)=f(x)^2$.
Take the derivative of both sides. $$f'(x^2) \cdot 2x = 2f(x) \cdot f'(x)$$ $$f'(x^2) \cdot x=f(x) \cdot f'(x)$$ 1) If $f'(x) = 0$, then $f(x) = C$, where $C$ is some real number.
Then from the functional equation we get $C = C^2$. Solving this equation we get $C=0$ or $C=1$. Then $f(x) = 0$ or $f(x) = 1$.

2) $f'(x) \ne 0$
Then we can get such a differential equation:
$$f(x)=\frac{f'(x^2) \cdot x}{f'(x)}$$