I am reading this paper https://www.math.ias.edu/~goresky/pdf/MatherBio.pdf from Mark Goresky. He talked about an example on page 5 saying that
The first example is an algebraic subset of Euclidean space such that no decomposition into smooth manifolds will be locally trivial in the $C^1$ sense. The variety $$xy(y − x)(y − zx) = 0 $$ consists of four ”sheets” meeting along the $z$ axis. Three of the sheets are simply a product with the z axis, but the fourth sheet twists around the axis. Any differentiable flow in the ambient Euclidan space, parallel to the $z$ axis, that preserves the first three sheets cannot preserve the fourth, because the derivative at any point on the $z$ axis is determined by the cross-ratio. So the homogeneity that is apparent in this example can only be realized by a continuous flow.
I don't understand what he means by saying that "the derivative at any point on the z axis is determined by the cross-ratio". Why there is no differentiable flow but only continous flow?