# Graph of the Dirichlet Function

we know, the Dirichlet function as: $$f(x) = \begin{cases} 1, &\text{if } x \text{ is irrational; and } \\ 0, &\text{if }x \text{ is rational}. \end{cases}$$ R.A. Silverman in his book Modern Calculus says that this wild function cannot be plotted at all while M.R.Spiegel in the book Advanced Calculus constructed a graph of $f(x)$ as two parallel lines with $x$-axe, gone via $1$ and $0$. The second athour says: The graph is shown in the adjoining Fig. 2-3. From its appearence it would seem that there are two functional values $0$ and $1$ corresponding to each value of $x$, i.e. that $f(x)$ is multiple-valued, whereas it is actually single-valued. What happens in these two point of views? Silverman would deny that this picture counts as "plotting" the function. What look like two solid lines are actually not solid at all: the top line contains only points whose $x$ coordinate is irrational, and the bottom line contains only points whose $x$ coordinate is rational. These facts are impossible to show in a picture.