For polynomials with non-negative integer coefficients and no constant term, the following simple (but not obvious) fact was observed by Paulina Szczuka.
Theorem. Each polynomial $f:\mathbb N\to\mathbb N$, $f:x\mapsto a_1x+a_2x^2+\dots+a_nx^n$, with integer coefficients and no constant term is continuous in the Golomb topology on $\mathbb N$.
Proof. Take any number $x\in\mathbb N$ and a basic neighborhood $f(x)+b\mathbb N_0$ of its image in the Golomb topology.
Since $f(x)$ is divisible by $x$, the number $b$ is relatively prime with $x$, so $x+b\mathbb N_0$ is a well-defined basic neighborhood of $x$ in $\mathbb G$. Observe that $$f(x+n\mathbb N_0)\subset f(x+b\mathbb Z)\subset f(x)+b\mathbb Z.$$
Since $f(\mathbb N)\subset\mathbb N$ and $f(0)=0$, the polynomial $f$ is not constant, so, for any $y\in Y$ in the finite set $Y:=\mathbb N\setminus (f(x)+b\mathbb N)$ the set $f^{-1}(y)$ is finite. The neighborhood $O_x=(x+b\mathbb N_0)\setminus \bigcup_{y\in Y}f^{-1}(y)$ of $x$ in the Golomb topology has the required property: $f(O_x)\subset f(x)+b\mathbb N_0$.
Corollary, The polynomial $f:\mathbb G\to\mathbb G$, $f:\mapsto x^3-12x^2+45x$ on the Golomb space is continuous but it is neither increasing nor decreasing (since $f(1)=34$, $f(3)=54$ and $f(5)=50$).
