Discontinuous functions from $\mathbb Q_p$ to $\mathbb R$

Question

The question is how we construct a function $f:\mathbb Q_p\to\mathbb R$ so that $f$ is discontinuous at every $x_0\in\mathbb Q_p$.

Answer 1

Let $D=\{0,1\}$ with the discrete topology, so that $D^\omega$ with the product topology is a Cantor set. $\mathbb{Q}_p$ is homeomorphic to $\omega\times D^\omega$ or, equivalently, to $D^\omega\setminus\{p\}$ for any $p\in D^\omega$. In particular, it has a countable dense subset $S$, and $\mathbb{Q}_p\setminus S$ is also dense in $\mathbb{Q}_p$. In fact it’s well-known that $\mathbb{Q}$ is dense in $\mathbb{Q}_p$, so we may take $S=\mathbb{Q}$, but any countable dense $S$ will work equally well.

Then $$\chi_S:\mathbb{Q}_p\to\mathbb{R}:x\mapsto\begin{cases}1,&x\in S\\ 0,&x\notin S\;, \end{cases}$$

the indicator (characteristic) function of $S$, is nowhere continuous, and in particular $\chi_\mathbb{Q}$ is nowhere continuous.