= Solution
{c}
For $r>0$, the <partial derivatives> $r_x=x/r$ and $r_y=y/r$ give
$$
\frac{\partial}{\partial x}\frac{x^2}{r}=\frac{2x}{r}-\frac{x^3}{r^3},\qquad
\frac{\partial}{\partial y}\frac{xy}{r}=\frac{x}{r}-\frac{xy^2}{r^3}.
$$
Adding these <partial derivatives> yields the <divergence>:
$$
\boxed{\nabla\cdot\mathbf f=\frac{3x}{r}-\frac{x(x^2+y^2)}{r^3}=\frac{2x}{\sqrt{x^2+y^2}},\qquad (x,y)\ne(0,0).}
$$
The original <vector field> is undefined at the origin. Since $\|\mathbf f\|=|x|\leq r$, it has the <continuous extension> $\mathbf f(0,0)=\mathbf0$. This extension is not <differentiable>: on $y=0$ its first component is $|x|$, whose <derivative> at zero does not exist. Thus the displayed <divergence> has no classical value at the origin.
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