The Golomb space $\mathbb G$ is the set $\mathbb N$ of positive integers endowed with the topology generated by the base consisting of arithmetic sequences $a+b\mathbb N_0:=\{a+bn:n\ge 0\}$ with $a,b$ relatively prime.
It is known that the Golomb space is topologically rigid, i.e., admits no non-trivial homeomorphisms. On the other hand, $\mathbb G$ has continuum many continuous self-maps. In particular, each polynomial $f:\mathbb G\to\mathbb G$, $f:x\mapsto a_1x+\dots+a_nx^n$, with integer coefficients and without the constant term is a continuous self-map of $\mathbb G$. But all known examples of non-constant continuous self-maps $f$ of the Golomb space have the property that for any $x\in\mathbb N$ each prime divisor $p$ of $x$ divides $f(x)$.
Question 1. Is there a non-constant continuous map $f:\mathbb G\to\mathbb G$ such that $f(x)=1$ for some $x\ne 1$?
We can also ask a more general
Question 2. Is it true that any map $f:F\to\mathbb G$ defined on a finite subset $F\subset\mathbb G$ extends to a continuous self-map $\bar f:\mathbb G\to\mathbb G$ of $\mathbb G$?
Remark. The last question is equivalent to asking if the set $C(\mathbb G)$ of all continuous self-maps of the Golomb space is dense in the space $\mathbb N^{\mathbb N}$ of all functions of $\mathbb N$, where $\mathbb N^{\mathbb N}$ is endowed with the Tychonoff product topology of discrete spaces $\mathbb N$.
