Give $Z$ the metric topology induced by the p-adic metric and we have defined a function $f:Z \to R$, where $f(z) = 1$ if $z \equiv 1 \mod p$ and $f(z) = 0$ otherwise. How would I go about proving or disproving the continuity of this function? Since $Z$ is given the metric topology, are $z \in Z$ just open balls with some distance given by the p-adic metric?
2026-04-03 01:05:34.1775178334
Proving/disproving continuity of function on p-adic topology
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The $p$-adic completion of $ a+p\mathbf{Z}$ is $ a+p\mathbf{Z}_p = \{ b \in \mathbf{Z}_p, |a-b|_p < 1\}$ and $ \mathbf{Z}_p = \bigcup_{a=0}^{p-1} a+p\mathbf{Z}_p$ where the union is disjoint.
$f$ is constant on each $a+p\mathbf{Z}$ thus it can be extended continuously to $a+p\mathbf{Z}_p$ and $\mathbf{Z}_p$