Vanishing prefactor at one time does not ensure a uniform Lipschitz bound ($f(x,y)=x|y{|}^{2/3}$)

A coefficient that vanishes at the initial time can hide failure of the uniform Lipschitz condition required for an initial value problem. For $f(x,y)=x|y{|}^{2/3}$, the map $y\mapsto f(0,y)$ is identically zero. But every rectangle of positive widths around $(0,0)$ contains some $x\ne 0$, andNo Lipschitz constant works uniformly on the rectangle. Actual uniqueness also fails: $y=0$ and $y=x^6/216$ are distinct differentiable solutions with initial value zero. Substitution gives $y^{\prime }=x^5/36=x|y{|}^{2/3}$ for the second solution. A condition at the single initial time cannot replace control on a neighbourhood.