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The inverse function theorem says that there exists an open set $U$ such that there is a local inverse of $f$ around a point $a$.
However, how would we actually find an open set containing this point $a$?
In particular, the function
$$f(x,y) = \begin{pmatrix} x^2 y + 2y - x \\ 3xy + 4x \end{pmatrix}$$ with the point $(0,0)$.

Twenty-six colours
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  • Hmm, I don't think so, the inverse function theorem holds when the determinant of the jacobian of $f$ is non-zero right? And it is in this case – Twenty-six colours Jun 07 '17 at 06:14
  • @Thomas Yes, but doesn't this case satisfy both requirements?
    The jacobian (differential) is non-zero (so has maximum rank) and also the function in question takes in two variables (a vector in $\mathbb{R}^2$) and it also gives out a vector in $\mathbb{R}^2$.     – Twenty-six colours Jun 08 '17 at 02:57
  • ups, my bad. Sorry – Thomas Jun 08 '17 at 14:50

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