Let $f: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}$ be sufficiently smooth with $f(x_0,y_0) = 0$ and the Jacobian $J_yf(x_0,y_0)$ of $f$ with respect to the second variable $y$ is invertible at a given point $(x_0,y_0)$. Then, the implicit function theorem states, that there exists a $\delta > 0$, such that with $B_\delta(x_0):=\{x: |x-x_0| < \delta\}$ there exists a differentiable function $g: B_\delta(x_0) \to \mathbb{R}^n$, such that it holds $$f(x,g(x)) = 0 .$$
My question is, how can I ensure, that $\delta$ has a at least a given size, i.e. I am looking for a lower bound for $\delta$, e.g. $$\delta > B(f,x_0,y_0),$$ where $B$ is some function that depends on $f,x_0,y_0$ (and probably on higher derivatives).
Thanks very much!