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Is there a software where I can graph $D(x) \cdot \sin(x)\ \vert \ D(x) \text{ Dirichlet's function}$?

The problem is that it's a known function, and I haven't seen in Wolfram|Alpha, Desmos and Geogebra that it's recognized, and I've also not seen there the option to define graph a function that's defined in cases.

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    $\begingroup$ The Dirichlet function is not very drawable. Even if you can plot it, you will not get much information from that picture. $\endgroup$
    – M. Winter
    Jun 30, 2017 at 11:51
  • $\begingroup$ In all those software you can do it. I just did it in MatLab, which is the one I had at hand. Define the function D(x) to take a floating point number. Inside it convert the number into a fraction, return the reciprocal of the denominator (I did it for this version of the Dirichlet function because it is prettier). Then create a vector for the $x$ axis. For example the numbers between 0 and 1 picked at a fine enough step. Evaluate $D(x)*\sin(x)$ at the points in the vector to get your vector of $y$. Then plot $x$ against $y$. In Matlab [N,D]=rat(x) gives you the fraction, plot(x,y) the plot. $\endgroup$
    – OR.
    Jun 30, 2017 at 12:07

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In the same sense that the graph of Dirichlet's function looks like the lines $y = 0$ and $y = 1$, with holes at every rational location in the lower line and holes at every irrational location in the upper, the graph $$ y = f(x) \chi_{\mathbf{Q}}(x) $$ looks like the graphs $y = 0$ and $y = f(x)$, with holes at every rational location in the line $y = 0$ and holes at every irrational location in the graph $y = f(x)$.

The product of sine and the Dirichlet function

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  • $\begingroup$ I was able to imagine that it looks like that, but I was wanting to draw it with some software. Off topic: Is it possible in pgfplots/TikZ to draw this, defining the function by cases? And, what's that $\chi_{\mathbb{Q}}(x)$ there? Is Dirichlet's function a characteristic function of $\mathbb{Q}$ or something? $\endgroup$
    – Gal Grünfeld
    Jun 30, 2017 at 11:53
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    $\begingroup$ 1. The graphical point is, you may as well just make two plots, $y = 0$ and $y = f(x)$, and assert that each has "gaps", with the gaps of one graph vertically aligned with the points of the other. 2. I don't know about pgfplots. 3. If by "Dirichlet's function" you meant $D(x) = 1$ if $x$ is rational and $D(x) = 0$ otherwise, then by definition yes, $D$ is the characteristic function of the rationals. :) $\endgroup$
    – Andrew D. Hwang
    Jun 30, 2017 at 12:03

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