A functional fix

Algebra Level 5

Suppose $f$ is a function from the positive integers to the positive integers that satisfies

$f(n) = \begin{cases} n - 600, & \text{ for } n > 2013,\\ f( f(n + 800)), & \text{ for } n \leq 2013. \\ \end{cases}$

How many fixed points does $f$ have?

Details and assumptions

A fixed point of a function $f$ is a value $x$ which satisfies $f(x) = x$ .