<s>[INST] zero Jacobian matrix determinant and local inverse Consider the mapping $f: \mathbb R^2 \backslash \{(0,0)\} \to \mathbb R^2$ given by
$$\begin{aligned}
        f(x,y) = \begin{pmatrix}
        (x^2-y^2)/(x^2+y^2) \\
        xy/(x^2+y^2)
        \end{pmatrix}
\end{aligned}$$
    Does $f$ have a local inverse at every point of $\mathbb R^2$?
Update: The interesting fact with this function is that we can not use the inverse function theorem to decide whether it is locally injective, since for all points $(x, y)$ we have $Jf (x, y) = 0$. Indeed,
\begin{align}
\det Jf(x,y) = & 
  \det \begin{pmatrix}
  \frac{\partial}{\partial x} \frac{ (x^2-y^2)}{(x^2+y^2)} 
  & 
  \frac{\partial}{\partial y} \frac{ (x^2-y^2)}{(x^2+y^2)}
  \\
  \frac{\partial}{\partial x} \frac{ xy}{(x^2+y^2)} 
  & 
  \frac{\partial}{\partial y} \frac{xy}{(x^2+y^2)}
  \end{pmatrix}
\\
=&
\det \begin{pmatrix}
  \frac{ 2x(x^2+y^2)-(x^2-y^2)2x}{(x^2+y^2)^2} 
  & 
  \frac{-2y(x^2+y^2)-(x^2-y^2)2y}{(x^2+y^2)^2}
  \\
   \frac{y(x^2+y^2)-xy(2x) }{(x^2+y^2)^2} 
  & 
  \frac{ x(x^2+y^2)-xy(2y) }{(x^2+y^2)^2} 
\end{pmatrix}
\\
=&
\det \begin{pmatrix}
  \frac{ 4xy^2}{(x^2+y^2)^2} 
  & 
  \frac{-4x^2y}{(x^2+y^2)^2}
  \\
   \frac{-x^2y+y^3 }{(x^2+y^2)^2} 
  & 
  \frac{ x^3-xy^2 }{(x^2+y^2)^2} 
\end{pmatrix}
\\
=&
 \frac{ 4xy}{(x^2+y^2)^2} 
\det \begin{pmatrix}
  y
  & 
 -x
  \\
   \frac{-x^2y+y^3 }{(x^2+y^2)^2} 
  & 
  \frac{ x^3-xy^2 }{(x^2+y^2)^2} 
\end{pmatrix}
\\
=&
 \frac{ 4xy}{(x^2+y^2)^2} \cdot \frac{ 1}{(x^2+y^2)^2} 
\det \begin{pmatrix}
  y
  & 
 -x
  \\
 {-x^2y+y^3 }
  & 
 { x^3-xy^2 }
\end{pmatrix}
\\
=&
 \frac{ 4xy}{(x^2+y^2)^2} \cdot \frac{ 1}{(x^2+y^2)^2} 
 [ y({ x^3-xy^2 }) +x({-x^2y+y^3 })]
\\
=&
 \frac{ 4xy}{(x^2+y^2)^2} \cdot \frac{ 1}{(x^2+y^2)^2} 
 [ x^3y-xy^3-x^3y+xy^3]
\\
=&
 \frac{ 4xy}{(x^2+y^2)^2} \cdot \frac{ 1}{(x^2+y^2)^2} 
 \cdot 0
\\
=&0
  \end{align}
 [/INST] Note that for any $z\in \mathbb R^2\setminus \{(0,0)\},$ $f(rz)=f(z)$ for all $r>0.$ This implies $f$ is constant on each ray from the origin. Such a function cannot be injective on any nonempty open set.
</s>