# 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?

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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.