Discussion
Consider the function \(F: \mathbb{R}^{2} \to \mathbb{R}^{2}\) given by
\[ F(x,y) = (x^{3} - 3xy^{2} - 3x,\ 3x^{2}y - y^{3} - 3y). \]
Then, for the function F, the inverse function theorem is
Answer: C
The inverse function theorem is not applicable at exactly two points.
With z = x + iy, the two components of F are the real and imaginary parts of \(h(z) = z^{3} - 3z\), since \(z^{3} = (x^{3} - 3xy^{2}) + i(3x^{2}y - y^{3})\). For a holomorphic map the Jacobian determinant is \(|h'(z)|^{2}\):
\[ \det J_{F} = |3z^{2} - 3|^{2} \]
which vanishes only at \(z = \pm 1\), the points (1, 0) and (-1, 0). Everywhere else the Jacobian is invertible and the theorem applies. So it fails at exactly two points; 'all points' ignores them, and one or three points miscount the roots of \(z^{2} = 1\).
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