(This is not a duplicate of another question on math.stackexchange, as that other question just basically asks for the answer to the question below, of which I have provided an answer to. My question is different.)
Whitney showed that for maps of two-manifolds into $\mathbb{R}^3$, a typical cross cap looks like the map $(x, y) \to (x, xy, y^2)$. Prove that this is an immersion except at the origin.
We compute the differential of the map $f(x, y) = (x, xy, y^2)$. This is precisely$$df = \begin{pmatrix} 1 & y & 0 \\ 0 & x & 2y\end{pmatrix}.$$We claim that this has maximal rank away from the origin. To see this, note that, if the second row is nonzero, then it will be linearly independent from the first row because it has zero in the first entry. Thus we simply need that $x \neq 0$ or $2y \neq 0$, which is true everywhere except the origin, as claimed.
My question though is, what does the image of the map $f$ look like? I have a poor time with visualizing stuff like this...
This is a famous surface, called the Whitney umbrella. It has the equation $y^2=x^2z$.