Polish Mathematical Olympiad, 2nd round, 1992/1993, problem 6

Finding the value at 500 of a function satisfying continuity condition with functional equation. We use the intermediate value theorem, which almost trivializes this problem. This problem comes from the Polish Mathematical Olympiad, 1992/1993.

[f-th]

I know it's not required, but what could the function be?
Obviously it isn't just 1/x, since f(1000) = 999.