Bounded partial derivatives give Lipschitz; a fractional power in $y$ alone does not

Let D be the rectangle $\{(x,y)\in\mathbb{R}^2\mid|x|\le1,|y|\le1\}$ and h and g are functions defined on D given by: $h(x,y)=xy^2$ and $g(x,y)=y^{2/3}$. Then: