$y^2 - x^3$ not an embedded submanifold How can I show that the cuspidal cubic $y^2 = x^3$ is not an embedded submanifold of $\Bbb{R}^2$? By embedded submanifold I mean a topological manifold in the subspace topology equipped with a smooth structure such that the inclusion of the curve into $\Bbb{R}^2$ is a smooth embedding. I don't even know where to start please help me. All the usual tricks I know of removing a point from a curve and see what happens don't work. How can I extract out information about the cusp to conclude it is not? Also can I put a smooth structure on it so it is an immersed submanifold? THankz.

It is better to view $y$ as the independent variable and $x=y^{2/3}$.  Since $2/3<1$, this has infinite slope at the origin for positive $y$ and infinite negative slope for negative $y$.  Hence the origin is not a smooth point of this graph, which is therefore not a submanifold.
