Exercise 4.6.2

To find $D_f$ consider $x\in D_f$ for the two cases $x\in A$ and $x\notin A$. If $x\in A$ then $f$ is not continuous, since $f(x)>0$ but for any $\delta >0$ we can find $y\in V_{\delta }(x)$ with $y\notin A$ (because $A$ is countable, $A^c$ must be dense) hence there is an unavoidable error of $|f(x)-f(y)|=f(x)>0$.

Now consider $x\notin A$, using the sequential criterion for continuity notice every sequence $(x_n)\to x$ has $f(x_n)\to 0$ (since if $x_n\in A$ converge to $0$ as $n\to \infty$, and $x_n\notin A$ are always $0$), now since $f(x)=0$ this shows $f$ is continuous at $x\notin A$.

Together we’ve shown $D_f=A$. Setting $A=Q$ and using a particular ordering recovers Thomae’s function. Hence we can view this as a generalization of Thomae’s function for arbitrary countable sets.