Could anyone advise me how to solve the following problem:
Find all $x \in \mathbb{R}$ such that $x-\lfloor x\rfloor= \dfrac{2}{\dfrac{1}{x} + \dfrac{1}{\lfloor x\rfloor}},$ where $\lfloor *\rfloor$ denotes the greatest integer function.
Here is my attempt:
Clearly, $x \not \in \mathbb{Z}.$
$\dfrac{x}{\lfloor x \rfloor} - \dfrac{\lfloor x \rfloor}{x}= 2$
$ \implies x^2 -2x\lfloor x \rfloor - \lfloor x \rfloor ^2 = 0$
$\implies x = (1 \pm \sqrt2 )\lfloor x \rfloor $
$\implies \{x\} = \pm\sqrt2 \lfloor x \rfloor$
Thank you.