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.