Example 1 Let positive real numbers $x, y$ satisfy $xy=1$. Find the range of the function
$$
f(x, y)=\frac{x+y}{[x][y]+[x]+[y]+1}
$$

(where $[x]$ denotes the greatest integer not exceeding $x$).
Solution: Without loss of generality, let $x \geqslant y$, then $x^{2} \geqslant 1, x \geqslant 1$.
(1) When $x=1$, $y=1$, at this time $f(x, y)=\frac{1}{2}$.
(2) When $x>1$, let $[x]=n,\{x\}=x-[x]=\alpha$, then $x=n+\alpha, 0 \leqslant \alphaa_{2}=a_{3}, a_{3}b_{2}>\cdots>b_{n}>\cdots .
\end{array}
$$

Thus, when $x>1$, the range of $f(x, y)$ is $\left[a_{2}, b_{1}\right)$, i.e., $\left[\frac{5}{6}, \frac{5}{4}\right)$.
In summary, the range of $f(x, y)$ is
$$
\left\{\frac{1}{2}\right\} \cup\left[\frac{5}{6}, \frac{5}{4}\right) \text {. }
$$

Note: This problem skillfully uses the monotonicity of the function $g(x)=x+\frac{1}{x}(x \geqslant 1)$.