Suppose that I have an implicit curve $$F(x, y)=0 $$ and a candidate $$ \overline{r} : I \rightarrow \mathbb{R}^2$$ for its parametrization.
$$ \overline{r}(t) =x(t)\hat{i}+y(t) \hat{j}, \ \ \ t\in I $$
How could I prove rigorously that the parametrization is correct, i.e. it attains all points of the curve and no extra points? I already know that $$ F(x(t), y(t))=0. $$
