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The problem from a book is:

Let $G\subset R^2$ be the graph of $g: R\rightarrow R, g(x)=|x|^{1/3}$. Show that G admits a smooth structure so that the inclusion $G\rightarrow R^2$ is smooth.Is it an immersion?

My question:

Obviously $G$ is not a regular submanifold of $R^2$, is it possible the inclusion $G\rightarrow R^2$ is smooth?

jizhou
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    I'm still waiting for help. There is a hint after the problem: "Consider the map f: R -> R given by f(t) = texp(-1/t) if t > 0; 0 if t == 0; texp(1/t) if t < 0". Does it suggest to use f(t) as a smooth structure of G? Concretely give G an atlas with single chart (G, inv(f)). Under this smooth structure, the smoothness of inclusion G->R^2 is actually the smoothness of: (f(t),g(f(t)). Am I right? – jizhou Apr 28 '16 at 02:22

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