# M. Spivak, problem 25 chapter 2

bigli
How can I show that the set {(x,|x|) , x in real numbers} is not the image of any immersion of R into R^2 ?

problem 25 chapter 2 differential geometry M. Spivak

christoff
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.

bigli
http://trainbit.com/files/0810149884/Emb_Submanifold.jpg

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HallsofIvy has told you exactly how to go about this. If there existed an immersion whose image was the set $\{(x,|x|)\}$, 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 $\{(x,|x|)\}$.