In this paper, we prove an extension of Mahler's theorem, a celebrated result of $p$-adic analysis. Mahler's original result states that a function from $N$ to $Z$ is uniformly continuous for the $p$-adic metric $d_p$ if and only if it can be uniformly approximated by polynomial functions. We prove the same result for functions from $A^*$ to $Z$, where $d_p$ is now the profinite metric defined by $p$-groups (pro-$p$ metric).
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