By (the proof) of Theorem 4.3 in this paper of Paulina Szczuka, each polynomial $f:\mathbb N\to\mathbb N$, $f:x\mapsto a_1x+a_2x^2+\dots+a_nx^n$, with integer coefficients is continuous in the Golomb topology on $\mathbb N$.
In particular, the polynomial $f(x)=x^3-12x^2+45x$ on the Golomb space is continuous but it is neither increasing nor decreasing (because $f(1)=34$, $f(3)=54$ and $f(5)=50$).