Is a locally constant function from a profinite abelian p-group constant modulo an open normal subgroup?

Question

Suppose $f$ is a p-adic valued function from a profinite abelian group $B$ ($Z_{p}$ or $Z_{p}^{*}$) such that for all $x$ in $B$ there is an open set $N(x)$ around $x$ such that $f$ is constant on $N(x)$. Then, my question is, for every such function does there exist an open normal subgroup $H$ such that $f$ is constant modulo $H$?