(In reply to re: Is my reasoning valid? Spoiler
Nice work, Harry.
Yes, this does pave the way for a proof by induction that f(x) = x for all positive x. Jer's method works fine, starting with x = 3. If f(3) = 3, then f(4) = 4 and f(5) = 5 and so on, and so on.
And that makes the problem solution = 4, just as Charlie said.
I am interested to hear about the behavior of f(x) when x is positive and non-integral, but at first blush my intuition says that f(x) = x for all x.