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

(where $[x]$ denotes the greatest integer less than or equal to $x$).
16. Suppose $x \geqslant y$, then $x^{2} \geqslant 1, x \geqslant 1$. There are two cases:
(1) When $x=1$, $y=1$, at this time $f(x, y)=\frac{1}{2}$.
(2) When $x>1$, let $[x]=n$, by definition and property (12) we can set $\{x\}=x-[x]=\alpha$, then $x=n+\alpha(0 \leqslant \alphaa_{2}=a_{3}, a_{3}b_{2}>\cdots>b_{n}>\cdots$. 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 $\left\{\frac{1}{2}\right\} \cup\left[\frac{5}{6}, \frac{5}{4}\right)$.