Exercise 2, § 2

In the field $\mathbf{Q}_p$ of $p$-adic numbers (Gen. Top., III, p. 322, exerc. 23) every $p$-adic integer $x \in \mathbf{Z}_p$ has one and only one expansion in the form $x = a_0 + a_1 p + \cdots + a_n p^n + \cdots$, where the $a_j$ are rational integers such that $0 \leq a_j \leq p - 1$ for each $j$. For each $z \in \mathbf{Z}_p$ put

show that, on $\mathbf{Z}_p$, $f$ is a continuous function which is not constant on a neighbourhood of any point yet has a zero derivative at every point.