Here is the graph of this function with some points highlighted as plus symbols for better view.

This function has interesting property: it's continuous at all irrational numbers. It's easy to see this if you notice that for any positive

*ε*there is finite number of points above the line

*y = ε*. That means for any irrational number

*x*you can always construct a

_{0}*δ*-neighbourhood that doesn't contain any point from the area above the line

*y = ε*.

To generate the data file with point coordinates I used Common Lisp program:

(defun rational-numbers (max-denominator)

(let ((result (list)))

(loop for q from 2 to max-denominator do

(loop for p from 1 to (1- q) do

(pushnew (/ p q) result)))

result))

(defun thomae-rational-points (abscissae)

(mapcar (lambda (x) (list x (/ 1 (denominator x)))) abscissae))

(defun thomae (max-denominator)

(let ((points (thomae-rational-points (rational-numbers max-denominator))))

(with-open-file (stream "thomae.dat" :direction :output)

(loop for point in points do

(format stream "~4$ ~4$~%" (first point) (second point))))))

(thomae 500)

To create the images I used gnuplot commands:

plot "thomae.dat" using 1:2 with dots

plot "thomae.dat" using 1:2 with points

and Photoshop.

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