The existence interval of the implicit function theorem

Question

When I learned the implicit function theorem. I met a question about the extension of the existence interval. Consider the function $F(x,y)=0,x\in\mathbb{R}^{+},y\in\mathbb{R}^{+}$ with $F(x_0,y_0)=0$. During appropriate assumptions, by using the implicit function theorem, we can get a function $y=y(x)$ near $(x_0,y_0)$, denote the neighborhood of $(x_0,y_0)$as $U_0$. Then near $\partial U_0$, we can use the implicit function theorem again and get a new neighborhood $U_1$, but the neighbirhoods may be more and more smaller. So the process may stop after a few steps. I want to know if we have good enough conditions, can the existence interval of $x$ be $[0,x_0]$?