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I read from several sources the statement that the set of points M[itex]\in[/itex]ℝ^{2}given by [itex](t, \, |t|^2)[/itex] is an example of differentiable manifold of class C^{1}but not C^{2}.

Is that true?

To be honest, that statement does not convince me completely, because in order for M to be a manifold, we should be able to find an atlas of charts [itex]x_i:U_i \rightarrow M[/itex] such that all the points of M are covered by the atlas.

So how do we cover the point (0,0) ?

Do we need to use 3 charts as follows?

[tex]x_1(t) = (t,\, t^2) \quad \, t\in(0,+\infty)[/tex]

[tex]x_2(t) = (t,\, -t^2) \quad \, t\in(-\infty, 0)[/tex]

[tex]x_3(t) = (t,t^2) \quad \, t\in(-1,1)[/tex]

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# Example of differentiable manifold of class C^1

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