一、(40 points) Let $[x]$ denote the greatest integer not exceeding the real number $x$. Suppose positive real numbers $x, y$ satisfy $x y=1$. Find the range of the function
$$
f(x, y)=\frac{x+y}{[x][y]+[x]+[y]+1}
$$
Let's assume $x \geqslant y$. Then $x^{2} \geqslant 1 \Rightarrow x \geqslant 1$.
(1) When $x=1$, $y=1$, at this time, $f(x, y)=\frac{1}{2}$.
(2) When $x>1$, let
$$
[x]=n,\{x\}=x-[x]=\alpha \text {. }
$$

Then $x=n+\alpha(0 \leqslant \alphaa_{2}=a_{3}, \\
a_{3}b_{2}>\cdots>b_{n}>\cdots .
\end{array}
$$

Therefore, when $x>1$, the range of $f(x, y)$ is $\left[a_{2}, b_{1}\right)$, which is
$$
\left[\frac{5}{6}, \frac{5}{4}\right) \text {. }
$$

In summary, the range of $f(x, y)$ is $\left\{\frac{1}{2}\right\} \cup\left[\frac{5}{6}, \frac{5}{4}\right)$.