If one has a parametric curve $(x(t), y(t))$ under what conditions is $y$ implicitly a function of $x$?

Question

If I have parametric curve $\left( x(t), y(t)\right)$ I want to know when it is the case that $y$ is implicitly a function of $x$. I don't believe the Implicit Function Theorem applies but I'm not knowledgable enough to know if this belief is correct. Is there an Implicit Function Theorem for parametrized curves?

Answer 1

If $x'(t_0)\ne0$, by the inverse function theorem, you can write $t=t(x)$ on an interval around $x_0=x(t_0)$. This gives you $y=(t(x))$ on that interval.