The figure-eight is the image of $f$, where $f(t)=(\cos t,\sin 2t),-\pi /2 <t<3\pi /2.$
How to use the definition of regular submanifold to deduce that the figure-eight is not a regular manifold of $\Bbb R^2$?
In Tu's book, it mentions that there exits a cross at the origin , and hence the figure eight is not regular.