On the Denjoy rank, the Kechris-Woodin rank and the Zalcwasser rank

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