M. Spivak, problem 25 chapter 2

As per the forum rules, nobody should be helping you with this unless you try working the problem first, and showing us where you got stuck. Moreover, it may help to cite the definition you are using for immersion when you begin.

 

Have you actually looked at the image of (x, |x|) in R₂ (that is, the graph of y= |x|). Can you see why it is NOT a smooth manifold? That should tell you what point to focus on in your proof.

 

HallsofIvy has told you exactly how to go about this. If there existed an immersion whose image was the set [itex]\{(x,|x|)\}[/itex], then this set would have a smooth manifold structure. So to prove that no immersion exists, it is sufficient to prove that that you cannot define a smooth manifold structure on [itex]\{(x,|x|)\}[/itex].

I know you said in your "attempt" at a proof that you didn't want to look at the point x=0, but my friend, this will be necessary, since this point is the reason no such immersion exists.