continuity - Find a discontinuous function defined on $mathbb{Q}$ embedded in $mathbb{R}$

I am reading Chapter 1 Example 11 of 'Counterexamples in Analysis' by Gelbaum and Olmstead. This section illustrates counterexamples of functions defined on $\mathbb{Q}$ embedded in $\mathbb{R}$ of statements that are usually true for functions defined on a real domain. Almost all examples have the assumption that the function (defined on a rational domain) is continuous, for example, the book gives a counterexample of:

A function continuous and bounded on a closed interval but not uniformly continuous.

My questions are, what is an example of a discontinuous real function defined on $\mathbb{Q}$, that is: $f:\mathbb{Q}\rightarrow\mathbb{R}$? Are all functions defined on $\mathbb{Q}$ discontinuous (similar to how functions defined on the set of natural numbers are always continuous)?

Answer

1). All functions defined on $\mathbb{N}$ are continuous (not discontinuous).

2). An example of a function $f: \mathbb{Q} \to \mathbb{R}$ that is discontinuous is $f = \chi_{\{0\}}$, i.e. $f(x) = 1$ iff $x = 0$ ($x \in \mathbb{Q}$). One can see that this is discontinuous by noting that $f(\frac{1}{n}) = 0$ for each $n \ge 1$, while $f(\lim_n \frac{1}{n}) = f(0) = 1$.