Jacobian and local invertibility of function Following is a question in the text book:
Show by an example that the condition that the Jacobian vanishes at a point is not necessary for a function to be locally invertible at that point..... and the answer given is  f : R$^2$- R$^2$ f(x,y) = (x$^3$, y$^3$)
I am a bit confused by the question itself because my understanding is that  for local invertibility we check that Jacobian  is not zero....also for f(x,y) = (x$^3$, y$^3$), J works out to 9x$^2$y$^2$. So dont we say that this function is not invertible at either x=0 or y=0 since J at these values becomes 0? I am unable to understand the solution given in the textbook
I request help to understand this problem and the given solution

It looks like a typo in the text. It should say "...the condition that the Jacobian does not vanish...".
$f(x,y)=(x^3,y^3)$ is invertible. Its inverse is $f^{-1}(x,y)=(\sqrt[3]{x},\sqrt[3]{y})$. What happens is that the inverse is not differentiable.
