I'm trying to improve my understanding of why the Newton-Kantorovich theorem is, in some ways, insufficient in the singular case.
Let $f:\mathbb{R}^n\to\mathbb{R}^n$, and let $Df$ denote its (Frechet) derivative. Suppose $f(x^*)=0$. Since the hypotheses of the Newton-Kantorovich theorem don't include any conditions on $Df(x^*)$, the theorem should be applicable even when $Df(x^*)$ is singular.
There seem to be challenges with this though. For instance, if $Df(x^*)$ is singular, we are not guaranteed a ball about $x^*$ in which $Df(x)$ is nonsingular. So we have to be more careful about how we pick our initial guess $x_0$. There is nothing in the theorem indicating what kind of regions contain valid initial guesses.
Evidently there are other strict conditions that hold if we try to apply the Newton-Kantorovich theorem to $f$ when $Df(x^*)$ is singular. In his 1980 thesis Analysis and Modification of Newton's Method at Singularitites, A.O. Griewank states the following (he seems to use $\nabla$ to denote the derivative):
If the [Newton-Kantorovich] theorem applies at $x_0$ with respect to the Euclidean norm of vectors and spectral norm of matrices, it can be seen that singularity of $\nabla f'(x^*)$ requires with $s:=x_0-x^*$, \begin{align} f(x^*+\lambda s)=f(x_0)(\lambda/||s||)^2. \end{align}
It does seem to severely restrict choices of $x_0$, but I don't understand why this is a necessary condition when the Newton-Kantorovich theorem holds at $x_0$ and $Df(x^*)$ is singular. Any hints or guidance would be appreciated.