Fixed points of Gauss Transformation

Question

Gauss Transformation defined as $\psi(x):[0,1]\rightarrow [0,1]$
defined as $\psi(x)= \frac{1}{x}-\lfloor \frac{1}{x}\rfloor, 0 < x\leq 1 $ and $0$ for $x=0$
I want to find fixed points for this transformation. Clearly $x=0$ is a fixed point. With the help of graph we can see infinite number of fixed points as its intersects with $y=x$ in infinite number if times.
if $x\neq 0$ then $\psi(x)=x$
$\frac{1}{x}-\lfloor \frac{1}{x}\rfloor = x$
$\frac{1}{x}-x=\lfloor \frac{1}{x}\rfloor $
How to solve this ?
The grpah is

Answer 1

Hint: Restrict $x$ to the interval $(1/(n+1), 1/n]$ to evaluate $\lfloor 1/x \rfloor$.