54. The function $f: R \rightarrow R$ is continuous. For each real $x$, the equality $f(x) \cdot f(f(x))=1$ holds. It is known that $f(1000)=999$. Find $f(500)$.
88.54. Answer. $f(500)=1 / 500$. Indeed, $f(999)=$ $=f(f(1000))=1 / f(1000)=1 / 999$. Therefore, the function takes the values 999 and $1 / 999$, which means there exists a number $a$ such that $f(a)=500$. But then $f(500)=f(f(a))=1 / f(a)=1 / 500$.