Recall that a set $S\subseteq\mathbb{R}^3$ is called a regular surface if for every $p\in S$, there exists an open neighborhood $B_p$ of $p$, and also an open set $A\subseteq\mathbb{R}^2$ and a function $f:A\to\mathbb{R}^3$ such that $f(A) = B_p\cap S$, $f$ is injective and continuously differentiable, and $Df$ is of rank 2.
I am asked to prove the very intuitive result that the solid unit ball is not a regular surface. Although ridiculously trivial result, the proof seems to be very hard and I have no idea where to start.