Suppose I have the iteration operator of the Newton method for some $\beta$-parameter dependent function $g_{\beta}: \mathbb{R} \rightarrow \mathbb{R}$. Let us assume that $g_\beta$ is in $C^{\infty}$ and in $C^{1}$ with respect to the parameter $\beta$. Call the corresponding Newton iteration operator $f_{\beta}$. Given an initial condition $z_0$ for which the Newton operator converges ($\forall \beta$). Then my questions is if the function
\begin{equation} F(\beta) := \lim_{n \rightarrow \infty } f^n_{\beta}(z_0) \end{equation}
is continuous with respect to $\beta$? I guess I would be able to find this in the literature somewhere. Unfortunately, I had no luck so far :(
Any help is welcome :)