Can a function $f:\mathbb{R}^2 \rightarrow \mathbb{R}^3$ have a derivative at $a$ with rank $<2$ and still have a tangent plane at $a$?
Solution 1:
Let $f(x,y) = (x^3,y^3,0)$. The image of $f$ is $\mathbb R^2 \times \{0\}$ which has itself as a tangent plane at all points. But $Df$ is $0$ at $(0,0)$.