Consider the following module $\mathcal{M}=(\mathbb{Z}_{p}^{*},\ \mathbb{Z}_{p-1},\cdot)$ in which the "additive" operation is defined by normal multiplication in $\mathbb{Z}_{p}^{*}$ and scalar "multiplication" is defined by exponentiation and reducing modulo $p-1$, i.e., $k*x=x^k$ and reducing $k$ modulo $p-1$, where $x\in\mathbb{Z}_{p}^{*}$ and $k\in \mathbb{Z}_{p-1}$. Now consider the topology on $\mathcal{M}$ induced by the metric $d$ which is defined by
$d(x,y)= \begin{cases} |ord_p(x)-ord_p(y)|, & \text{if $ord_p(x) \neq ord_p(y)$} \\ 1, & \text{if $ord_p{x}=ord_{p}(y)$ and $x\neq y$}\\ 0, & \text{$x=y$} \end{cases}.$
(Notice that the topology generated by this metric is the discrete topology.)
I'm trying to prove that addition in this module is continuous with respect to this topology. I've attempted to prove continuity using by proving that it's continuous at an arbitrary point. So far I have the following:
Case 1) Let $ord_p(x) \neq ord_p(a)$ and $ord_p(y) \neq ord_p(b)$.
Let $(x,y), (a,b) \in\mathbb{Z}_{p}^{*} \times \mathbb{Z}_{p}^{*}$ with $d((x,y),(a,b)) \lt \delta$ for some $\delta \gt 0$ and let $\epsilon \gt 0$. Then $ \sqrt{ (ord_p(x)-ord_p(a))^2 + (ord_p(y)-ord_p(b))^2} \lt \delta$. From here we can obtain that $|ord_p(x)-ord_p(a)|\lt \delta$ and $|ord_p(y)-ord_p(b)| \lt \delta.$ Now let $x \cdot y = z$ and $a \cdot b = c$ for $z,c \in \mathbb{Z}_{p}^{*}$. Then $|ord_p(z)-ord_p(c)|= |ord_p(x\cdot y) - ord_p(a \cdot b)|$.
This is where I get stuck. I have no idea how to proceed from here. Am I going about this the correct way, and if so, how do I continue from this point?