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Suppose I have a smooth curve $\gamma: \mathbb{R} \rightarrow \mathbb{R}^2$ in the $xy$-plane given by $t \mapsto \gamma(t)=(\gamma_1(t),\gamma_2(t))$ which intersects the $x$-axis transversely. Is it then possible to locally express $\gamma$ in terms of $\gamma_2$?

I have not been able to construct a counter example yet but I have not been able to come up with a proof either. Any help is welcome.

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    The implicit function theorem should help you out here.2012-08-08

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