Solution
= Solution
At a point with $xy\ne0$, the <determinant> of the <derivative> is
$$
\det Df(x,y)=\frac{5y^4}{3x^{2/3}}\ne0.
$$
The <inverse function theorem> therefore supplies neighbourhoods on which $f$ has a continuously differentiable, hence <differentiable>, inverse. Concretely, if $(u,v)=f(x,y)$, the local inverse is the restriction of
$$
\boxed{y=v^{1/5},\qquad x=(u-v^{2/5})^3},
$$
which is continuously differentiable near the image whenever $y\ne0$.