If directional derivatives coincide, does one lose injectivity?

Question

Suppose you have a smooth function $\sigma = (\sigma_1,\sigma_2,\sigma_3):\mathbb{R}^2\rightarrow\mathbb{R}^3$. Write $\sigma_x:=\left(\frac{\partial\sigma_1}{\partial x},\frac{\partial\sigma_2}{\partial x},\frac{\partial\sigma_3}{\partial x}\right)$ and similarly for $\sigma_y$. If at some $p\in\mathbb{R}^2$ we have $\sigma_x = \sigma_y$, then is it still possible for $\sigma$ to be a bijection onto its image?

Answer 1

Sure. For instance, let $\sigma(x,y)=(x^3,y^3,0)$. Then $\sigma_x(0,0)=\sigma_y(0,0)=(0,0,0)$, but $\sigma$ is a smooth injection.