Give an example of:
1)$f:\mathbb R^2 \to \mathbb R^2$ such that $f$ is invertible in some neighbourhood of $x_0$ (that is $f$ is locally invertible) but $|Jf(x)|=0$ (jacobian determinant) $\forall x$ in the neighbourhood of $x_0$
2)$f: \mathbb R^2 \to \mathbb R^2$ such that $f$ is invertible and differentiable in some neighbourhood of $x_0$, continuous differentiable in $x_0$ but $J|f(x)|=0$ $\forall x$ in the neighbourhood of $x_0$
3)$f: \mathbb R^2 \to \mathbb R^2$ such that $f$ is differentiable in the neighbourhood of $x_0$, $|Jf(x)|\neq 0$ $\forall x$ in the neighbourhood of $x_0$ but $f$ is not invertible in any neighboourhood of $x$ $\forall x\in V_\alpha({x_0})$ (neighbourhood of $x_0$)
Unfortunately I haven´t been able to give examples of such fucntions; I think that this functions are "very pathological",and I would really appreciate if you can help me with this problem