Btukfyl

Could someone please give me guidance on how to plot this special function? I honestly have no idea where even to start (I've never actually done a tikzpicture from scratch).

[

f(x) = left{begin{array}{lr}

    0, &x  text{irrational}\

    frac{1}{q}, &x = frac{p}{q}   text{in lowest form}\[10pt]

    end{array}right}

]

It should look something like this:

Here's a sagetex solution using the computer algebra system, SAGE, to do the computations.

documentclass[border=3pt]{standalone}

usepackage{sagetex}

usepackage{pgfplots}

pgfplotsset{compat=1.15}

begin{document}

begin{sagesilent}

xvalue = 

yvalue = 

for q in range(1,101):

    for p in range(1,q):

        xvalue += [(p*1.0/q*1.).n(digits=2)]

        yvalue += [(1/(q/gcd(p,q))).n(digits=2)]

output = r""

output += r"begin{tikzpicture}"

output += r"begin{axis}["

output += r"title={Thomae's function},"

output += r"xlabel=$x$,"

output += r"ylabel=$f(x)$,"

output += r"]"

output += r"addplot[only marks,mark options={mark size=.5pt}] coordinates {"

for i in range(0,len(xvalue)):

    output += r"(%s, %s)"%(xvalue[i],yvalue[i])

output += r"};"

output += r"end{axis}"

output += r"end{tikzpicture}"

end{sagesilent}

sagestr{output}

end{document}

The result running in Cocalc is shown below:

By changing the value of q from 101 to something more, you'll get more points. SAGE is not part of the LaTeX distribution, so you would need to install it on your computer or, to avoid that, get a free Cocalc account. The documentation for SAGE is here and the documentation for sagetex is here on CTAN. The SAGE programming involves a mixture of SAGE commands mixed into Python code. If your work involves mathematics, then this is a very convenient package to learn.

Welcome to TeX.SE! Here is a proposal. Of course, this is just a cartoon because I had to cut off q, as is done in your screen shot.

documentclass[tikz,border=3.14mm]{standalone}

begin{document}

begin{tikzpicture}[scale=8]

draw [-stealth] (-0.1,0) -- (1.1,0);

draw [-stealth] (0,-0.1) -- (0,1.1);

foreach X in {1,...,7}

{ifnumX=1

draw (0.02,1/X) -- (-0.02,1/X) node[left]{$1$};

else

draw (0.02,1/X) -- (-0.02,1/X) node[left,xshift={(-(1+pow(-1,X)))*3pt}]{$frac{1}{X}$};

fi

}

foreach X [evaluate=X as Ymax using {int(X-1)}]in {5,4,...,2}

{foreach Y in {1,...,Ymax}

 {draw (Y/X,0.02) -- (Y/X,-0.02) node[below,fill=white]{$frac{Y}{X}$};

 fill ({Y/X},1/X) circle(0.2pt); 

 }

}

end{tikzpicture}

end{document}

ADDENDUM: Without external programs, compiles in below 1sec on a 4 years old MacBook pro.

documentclass[tikz,border=3.14mm]{standalone}

begin{document}

begin{tikzpicture}[scale=8]

draw [-stealth] (-0.1,0) -- (1.1,0);

draw [-stealth] (0,-0.1) -- (0,0.6);

foreach X in {1,...,7}

{ifnumX=1

else

draw (0.02,1/X) -- (-0.02,1/X) node[left,xshift={(-(1+pow(-1,X)))*3pt}]{$frac{1}{X}$};

fi

}

foreach X [evaluate=X as Ymax using {int(X-1)}]in {25,24,...,2}

{foreach Y in {1,...,Ymax}

 {ifnumX<6

 draw (Y/X,0.02) -- (Y/X,-0.02) node[below,fill=white]{$frac{Y}{X}$};

 else

 draw[ultra thin] (Y/X,0.01) -- (Y/X,-0.01);

 fi

 fill ({Y/X},1/X) circle(0.2pt); 

 }

}

foreach X in {0,1,...,80}

{fill (X/80,0) circle(0.2pt); }

end{tikzpicture}

end{document}