On the denjoy rank, the kechris-woodin rank and the zalcwasser rank

We show that the Denjoy rank and the Zalcvasser rank are incomparable. We construct for any countable ordinal α differentiate functions f and g such that the Zalcwasser rank and the Kechris-Woodin rank of f are α + 1 but the Denjoy rank of f is 2 and the Denjoy rank and the KechrisWoodin rank of g are α + 1 but the Zalcwasser rank of g is 1. We then derive a theorem that shows the surprising behavior of the Denjoy rank, the Kechris-Woodin rank and the Zalcwasser rank.