In Numerical Analysis, the problem of finding an approximation for a root of an equation $$ f(x)=0 $$ may be solved using Fixed Point Iteration (refer to https://en.wikipedia.org/wiki/Fixed-point_iteration ). To use this method, one must write the previous equation as $$ x=F(x) $$ ($F$ is called an iteration function) and, if $f$ (which can be assumed to be differentiable) has a single root $\xi$ around an interval $(a,b)$, the sequence $$ x_{k+1}=F(x_k),k=0,1,2,... $$ converges to $\xi$ as long as $|F'(x)|<1$ in a small interval around $\xi$ in which the initial step $x_0$ is taken.
Now, given some equation $f(x)=0$, one can find several ways to write it as $x=F(x)$. As I discussed this matter with some of my students, I wondered if there was a way to guarantee that there is always a suitable $F$ (that is, $|F_1(x)|<1$ around $\xi$), given any $f(x)=0$. Of course, if there was some example in which this is impossible, then the question would also be answered.