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 $\alpha$ differentiable functions $f$ and $g$ such that the Zalcwasser rank and the Kechris-Woodin rank of $f$ are $\alpha +1$ but the Denjoy rank of $f$ is 2 and the Denjoy rank and the Kechris-Woodin rank of $g$ are $\alpha +1$ 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.