1) Is it possible that be given a suitable smooth atlas to the set $M:=\{(x,y)\in\mathbb{R}^2: y=|x|\}$ so that $M$ to be a differentiable manifold? Why? How?

2) How can I prove that M is not an embedded smooth submanifold of $\mathbb{R}^2$?

In really, How can I show that $M$ is not the image of any $C^{\infty}$- immersion of $\mathbb{R}$ into $\mathbb{R}^2$?