Functional equation $f(a)·f(b)=\Bigl\lbrace\frac{1}{ab}\Bigr\rbrace$ I am trying to get all possible solutions of the following functional equation:
$$f(a)·f(b)= \Bigl\lbrace\frac{1}{ab}\Bigr\rbrace$$
Where {} mean fractional part function.
Solutions only need to be valid inside open interval $a,b \in (0,1)$.
I am not really fluent in functional equations, any help would be useful.

We have
$$\Biggl(f\left(\frac{1}{2}\right)\Biggr)^2=\left\{\frac{1}{\left(\frac{1}{2}\right)^2}\right\}=\{4\}=0\,,$$
so
$$f\left(\frac{1}{2}\right)=0\,.$$
Consequently, for $a\in(0,1)$, we get
$$0=f(a)\,f\left(\frac{1}{2}\right)=\left\{\frac{1}{a\cdot\left(\frac12\right)}\right\}=\left\{\frac{2}{a}\right\}\,.$$
However, this is not true, for example, when $a=\frac{3}{4}$.  Hence, there is no such function $f:(0,1)\to\mathbb{R}$.
