In my studies, I'm using the definition of a manifold that is a set $V\subset R^n$ such that for any $x\in V$ occurs any of the itens in the theorem below.
This said, my difficult is to show that the set $V=\{(x,y)\in R^2 | y=-x^2, $ for $ x \leq0; y=x^2 $ for $x\geq0\}$ IS NOT a $C^\infty$ submanifold in $R^2$ of dimension 1. I already see that this set is the union of two parts of parabolas, and I've tried to show this by contradiction: Suppose that there is a $C^\infty$ embedding $\phi(t)=(x(t),y(t))$ with its image on the set $V$ defined in an open neighborhood of $0$ such that $\phi(0)=0$, and my attempt was to show that $\phi^\prime(0)=0$ which is a contradiction with the fact that it is an embedding, but it was fruitless. Another difficult that I'm having is to show that the set $K=\{(x,y)\in R^2 | \sqrt(x^2+y^2) \leq 1 \}$ is not a manifold. I know that the problem is that $K$ has boundary, but I don't know how to argue. Any hint?
