Good day everyone !
I'm just curious on how to plot (preferable in matlab or mathematica) the following stepwise function.
$ f(x) := \begin{cases} \dfrac{1}{q}~~~:~x=\dfrac{p}{q} \in \mathbb{Q} ~~\text{(in reduced form)}\\ 0~~~:~x \not \in \mathbb{Q} \end{cases} $
to illustrate that it is continuous at every irrational in the interval $(0,1)$ but discontinuous at every rational in $(0,1)$. Thanks in advance.
On
Before giving an answer, I first define the term "plot". Given small $\epsilon >0$, the plot of a function $f : \mathcal{D} \to \mathbb{R}$ is the set $\mathcal{R}_f = \{ B_{\epsilon}\big((x,f(x))\big) \mid x \in \mathcal{D} \} \subset \mathbb{R}^2$ where $\mathcal{D} \subseteq \mathbb{R}$ is the domain of $f$. Using this definition, observe that the plot of the function $f(x) = 0$ if $x\in \mathbb{Q}$ and $f(x)=1$ if $x\in \mathbb{Q}^{c}$ will be two horizontal cylinders $\mathbb{R}\times (1-\epsilon,1+\epsilon)$ and $\mathbb{R}\times (-\epsilon,\epsilon)$. Therefore, as $\epsilon$ approaches zero, it represents the lines $y=1$ and $y=0$. Now try to plot your function using this idea.
Plotting completely is not possible… How to represent an infinite number of discontinuity points?
Below is an approximate picture from my website.