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Suppose I have the curve $\gamma: \mathbb{R} \rightarrow \mathbb{R}^2$ given by $\gamma: t \mapsto (\gamma_1(t),\gamma_2(t)) =(t^2,t)$. If I want to apply the implicit function theorem to this to see if $\gamma_1$ can be expressed in $\gamma_2$ at $t=0$ then I need to show that $d \gamma_2 / d \gamma_1$ is non-zero. However, $d \gamma_2 / d \gamma_1 \rightarrow \infty $ for $t \rightarrow 0$. So in these cases you cannot apply the implicit function theorem?

Or can I just compactify the plane by "adding the point at infinity" and then apply the implicit function theorem.

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Strange way use the implicit function theorem, given you don't have a $F(x,y)=0$. The tangent vector at $t=0$ is vertical, so you can express $x=x(y)=y^2$, but not $y=y(x)$. – enzotib Aug 9 '12 at 18:53
up vote 4 down vote accepted

I don't see an implicit function here. You'd have an application of the implicit function theorem if you described the curve implicitly as $f(x,y) = x - y^2 = 0$. Then what you want in order to have $x$ expressible locally as a function of $y$ is $\partial f/\partial x \ne 0$, which it is.

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